2nd TLL Workshop

To achieve high degree of squeezing, an optical cavity is often employed to enhance the interaction time between light and matter. Here, we propose to utilize the effect of coherent population trapping (CPT) to generate squeezed light without any optical cavity [1]. Due to the slow propagation speed of light arising from the CPT effect, a medium with a large optical density (OD) can also result in a long light-matter interaction time. Consequently, coherent light passing through such high-OD CPT medium can make its variance highly squeezed at some quadrature angle even in a single passage. Our study showed that the squeezing of 11 dB can be achieved with an OD of 1,000 which is accessible by the current technology. A larger OD can further increase the degree of squeezing. As the light intensity and the two-photon detuning are the key factors in the CPT nonlinearity, we demonstrated that the minimum variance at a given OD can be reached with wide ranges of these two factors, showing the proposed scheme is flexible and robust. This work opens a new avenue in the generation of squeezed light and can advance the technologies in quantum optics and in quantum information manipulation with continuous variables. This work was supported by the Ministry of Science and Technology, Taiwan under the grant numbers of MOST 105-2923-M-007-002-MY3 and MOST 105-2119-M-007-004.

1. You-Lin Chuang, Ray-Kuang Lee, and Ite A. Yu, arXiv:1703.09890 (2017).

Over the recent years, negatively charged nitrogen-vacancy centers (NV) in diamond have emerged as a promising future technology for sensing applications. That is due to their high sensitivity and inherent scalability, which ranges from single NV centers, offering a few nanometre spatial resolution, to bulk sensors, providing enhanced field sensitivity. Despite these advantages, however, several important fundamental and technological challenges of NV-based sensing remain open.

Recent developments in magnetic field sensing with negatively charged nitrogen-vacancy centers in diamond employ magnetic-field dependent features in the photoluminescence. These features can be studied in presence of the microwave radiation or this radiation can be eliminated [1].

In the present study in addition of the laser radiation which polarized the NV centers in the ground state, we employ microwave field in different frequency ranges. When the microwave field frequency is in the range of 5,9 GHz and the magnetic field around 0.1 T is applied, we are studying microwave radiation caused resonant depolarisation of the NV centres in the ground state due to transitions between mixed levels in the vicinity of anti-crossing point and distant spin state magnetic sub-level. Dependence of the contrast of optically detected magnetic resonances and width of these resonances on the strength of the magnetic field is measured. Measured signals are compared with a numerical model that accounts for the hyperfine interaction of NV electronic angular momentum with the nitrogen nuclear spin and magnetic field.

If the microwave field frequency is much lower, below 100 MHz, but strength of the external magnetic field still is around 0.1 T, we are studying NV centre angular momentum resonant depolarisation by the microwave radiation causing transitions between magnetic sub-levels in the vicinity of the anti-crossing point. Again, the width and contrast of these signals as they depend on the external magnetic field, are measured. The experimental results are compared with the numerical model.

Finally, the effect of cross relaxation between NV centres and substitutional nitrogen [2] on the optically detected magnetic resonances in the vicinity of the magnetic field of 0.05 T – their contrast and width, is studied.

All the studies are supported by the model of NV centre interaction with light in the presence of magnetic field and microwave radiation, based on our previous experience of similar studies in atomic systems [3].

1. Huijie Zheng et. al arXiv:1701.06838v1, 24 Jan 2017
2. L. T. Hall et al Nature Communications, 5 January 2016
3. M. Auzinsh, D. Budker, S. Rochester, Optically Polarised Atoms, Oxford University Press, 2014.

We investigate the topological properties of spinless fermionic polar molecules loaded in a multi-layer structure with electric dipole moment polarized to the normal direction by an external electric field. We show that when fermions are paired between neighbouring layers at zero temperature, unpaired Majorana Zero Modes (MZMs) appears in the top and bottom layers near Fermi energy. The resulting topological state is of BDI class with non-trivial winding number index in $Z$, and its stability is protected by the time reversal symmetry. We further calculate the entanglement entropy, entanglement spectrum, and the critical temperature for realizing such MZMs, and then discuss the observation of such 2D quantum degenerate MZMs in the time-of-flight experiments.

Atoms excited to high-lying Rydberg states with a principal quantum number above 50 have recently attracted a significant attention [1]. The strong interaction between the Rydberg atoms allows one to investigate non-linear quantum optics at the level of individual quanta. This is achieved by coherently coupling slowly propagating photons to strongly interacting atomic Rydberg states under the conditions of the electromagnetically induced transparency (EIT) [2]. In an usual Rydberg EIT, a ladder atom-light coupling configuration is typically employed involving an atomic ground state, an intermediate excited state and a Rydberg state. Here we propose to use for Rydberg EIT a more complicated double tripod level scheme, shown in Fig. 1. In the double tripod scheme two probe laser fields are propagating inside the atomic medium leading to a two component (spinor) slow light. In the case of non-interacting atoms the propagation of the two-component slow light has been recently demonstrated experimentally [3]. In comparison to previously used schemes for quantum nonlinear optics with Rydberg states, the double tripod scheme can combine spin-orbit coupling for the spinor slow light with an interaction between photons. In a ladder atom-light coupling configuration the interaction is always attractive independent from the detuning [4]. In contrast, in the proposed scheme the interaction can become repulsive if the one-photon detunings have opposite signs.

Fig. 1. Double tripod atom-light coupling scheme involving the Rydberg levels s₁ and s₂ .

1. M. Saffman et al., Rev. Mod. Phys. 82, 2313 (2010).
2. D. Petrosyan et al., Phys. Rev. Lett. 107, 213601 (2011).
3. M.-J. Lee et al., Nat. Commun. 5, 5542 (2014).
4. O. Firstenberg et al., Nature 502, 71 (2013).

Usually optical lattices are produced by interfering a number of laser beams. The atoms are trapped at the intensity minima or maxima of an emerging interference pattern. The tunneling matrix elements for atoms in such lattices are real, so the atomic motion is not affected by a magnetic flux. On the other hand, the magnetic flux can be induced via the laser-induced tunneling between the lattice sites or the lattice shaking.

Here we consider alternative methods of producing topological optical lattices. In the first part of the talk we shall present a novel way of creating an optical lattice affected by a non-staggered magnetic flux without any conventional optical lattice added. Two atomic internal states are involved and their energies have opposite gradients in one spatial direction. The states are coupled by a pair of multi-frequency laser beams counter-propagating in a direction perpendicular to the energy gradient. Such a multi-frequency perturbation together with the energy gradient effectively creates a square optical lattice affected by a non-staggered magnetic flux. The energy bands of the lattice can be characterized by unit Chern numbers [1].

Subsequently we shall discuss semi-synthetic optical lattices involving atomic internal states as sites in an extra dimension. By taking a standard 1D optical lattice and adding the extra dimension obtained by laser-assisted transitions between atomic long lived states, one can effectively engineer a 2D lattice involving both real and synthetic dimensions, and the lattice is affected by a non-staggered magnetic flux [2]. In the initial treatments such semi-synthetic lattices has a square geometry. Here we also talk about recent analysis of semi-synthetic optical lattices with a non-square geometry [3,4].

1. T. Andrijauskas, I. B. Spielman and G. Juzeliunas, arXiv:1705.11101.
2. A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B. Spielman, G. Juzeliūnas, and M. Lewenstein, Phys. Rev. Lett. 112, 043001 (2014).
3. E. Anisimovas, M. Račiūnas, C. Sträter, A. Eckardt, I. B. Spielman, G. Juzeliūnas, Phys. Rev. A 94, 063632 (2016).
4. D. Suszalski and J. Zakrzewski, Phys. Rev. A 94, 033602 (2016).

Rydberg atoms [1] are characterized by large separation between the electron and the ion core, which leads to strong polarizability and extreme long-range dipole-dipole (DD) interactions.

In the phenomenon of dipole blockade [2], the DD interaction causes the applied laser fields to be off-resonant, such that only a single atom can be excited within the “blockade sphere” [3], while simultaneous excitation of two/multiple Rydberg atoms will be suppressed.

In our work, we aim for the best experimental parameters necessary to achieve a large (around 50 mm ) blockade radius.

Available in the literature data [4] for the strongest Förster resonances in ⁸⁷Rb, with equal or different principal quantum numbers for the two atoms, alows for maximum blockade radius of 10.02 mm for the 58d_3/2+58d_3/2→60p_1/2+56f_5/2 and 20.03 mm for the 80s_1/2+83s_1/2→80p_1/2+82p_1/2 transitions. Following the procedure outlined in [5], we are currently searching for better candidates among the multiple possible Förster resonances, which will enable us to reach the desired 50 mm radius.

Simultaneously, basing our calculations on [6], we are deriving a general expression for the angular dependence of the DD interaction strength for Förster resonance transitions, where the initial and final states have arbitrary mj-sublevels. The generalized expression should be further summed over all possible Zeeman components of the initial and final states, in which we plan to implement analytical, as well as numerical techniques.

This work was supported by the Trilateral grant of the Latvian, Lithuanian, and Taiwanese Research Councils FP-20338-ZF-N-100.

References

1. T. F. Gallagher, Rydberg Atoms, (Cambridge University Press, Cambridge, England, 1994).
2. M. D. Lukin et al., Phys. Rev. Lett. 87, 037901 (2001); A. Gaetan et al., Nature Phys. 5, 115 (2009).
3. D. Tong et al., Phys. Rev. Lett. 93(6), 063001 (2004).
4. T. Förster, Ann. Phys. 437, 55 (1948).
5. T. G. Walker and M. Saffman, Phys. Rev. A 77, 032723 (2008); I. I. Beterov and M. Saffman, Phys. Rev. A 92, 042710 (2015).
6. R. Faoro, Ph.D. Thesis, Université Paris-Saclay, 2015; Ch. Tresp, Ph.D. Thesis, Universität Stuttgart, 2017.

Three-body Förster resonances at long-range interactions of Rydberg atoms were first predicted and observed in Cs Rydberg atoms [1]. In these resonances, one of the atoms carries away an energy excess preventing the two-body resonance, leading thus to a Borromean type of Förster energy transfer. The experiment in [1] was done with an ensemble of ~10⁵ Cs atoms in an interaction volume of ~200 µm in size. Therefore, the three-body Förster resonance was in fact observed as the average signal for the large number of atoms N>>1.

In this report we present the first experimental observation of the three-body Förster resonance Rb(nP_3/2) + Rb(nP_3/2) → Rb(nS_1/2) + Rb([n+1]S_1/2) for a few Rb Rydberg atoms with n=36, 37. In our experiment, N=2-5 Rydberg atoms in the initial nP_3/2 Rydberg state interact in a single volume of ~20 µm in size. This volume is formed at the intersection of the two tightly focused laser beams that excite Rydberg states at the center of the cold Rb atom cloud in a magneto-optical trap [2]. Using the selective field ionization technique with the detection efficiency of 70%, the measured Förster resonance spectra are post-selected over the number of the detected Rydberg atoms N=1-5 [3] and then additionally processed to extract the true multi-atom spectra taking into account finite detection efficiency. The spectra represent the measured dependence of the fraction of the atoms in the final nS_1/2 state on the applied dc electric field, which controls the Förster resonance detuning, for various N.

Figure 1 shows the Stark-tuned Förster resonance in Rb atoms observed for various numbers of the interacting atoms N=2-5. In Fig. 1(a) the atoms are in the initial state 37P_3/2(|M_J|=1/2). The main peak at 1.79 V/cm is the ordinary two-body resonance that occurs for all N=2-5 [2-4]. The additional peak at 1.71 V/cm is the three-body resonance that is absent for N=2 and appears only for N=3-5.

The feature at 1.71 V/cm can in principle be caused by the imperfection of the electric-field pulses used to control the Förster resonance [4]. In order to check for this effect, the resonance has also been recorded for the atoms in the initial state 37P_3/2(|M_J|=3/2), as shown in Fig. 1(b). Here we see again that the main peak at 2.0 V/cm is the ordinary two-body resonance that occurs for all N=2-5. The additional peak at 2.14 V/cm is the three-body resonance that is absent for N=2 and appears only for N=3-5. We conclude that the three-body resonances really take place, as their positions and behavior well agree with theoretical calculations.

As a result, we have found clear evidence that there is no signature of the three-body Förster resonances for exactly two interacting Rydberg atoms, while it is present for the larger number of atoms. As the observed three-body resonance appears at the different dc electric field with respect to the two-body resonance (the difference increases for the lower n [1], and we have checked for it at n=36), it represents an effective three-body operator, which can be used to directly control the three-body interactions. This can be especially useful in quantum simulations and quantum information processing with neutral atoms in optical lattices.

This work was supported by the RFBR Grants No. 16-02-00383 and 17-02-00987, the Russian Science Foundation Grant No. 16-12-00028 (for laser excitation of Rydberg states), the Siberian Branch of RAS, the Novosibirsk State University, the public Grant CYRAQS from Labex PALM (ANR-10-LABX-0039) and the EU H2020 FET Proactive project RySQ (Grant No. 640378).

Fig. 1. Stark-tuned Förster resonance in Rb atoms observed for various numbers of atoms N=2-5: (a) atoms are in the initial state 37P_3/2(|M_J|=1/2); (b) atoms are in the initial state 37P_3/2(|M_J|=3/2). The main peaks are 2-body resonances, the additional peaks are 3-body resonances.

References

1. K. M. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J. Piotrowicz, A. W. Carr et al., Phys. Rev. A 92, 022336 (2015).
2. S. Ravets, H. Labuhn, D. Barredo, L. Beguin, T. Lahaye, and A. Browaeys, Nat. Phys. 10, 914 (2014).
3. I. I. Beterov, M. Saffman, E. A. Yakshina, D. B. Tretyakov, V. M. Entin, S. Bergamini, E. A. Kuznetsova, and I. I. Ryabtsev, Phys. Rev. A 94, 062307 (2016).
4. E. A. Yakshina, D. B. Tretyakov, I. I. Beterov, V. M. Entin, C. Andreeva, A. Cinins, A. Markovski, Z. Iftikhar, A. Ekers, and I. I. Ryabtsev, Phys. Rev. A 94, 043417 (2016).

Floquet engineering is a form of quantum engineering based on a periodic external driving of a controllable quantum system. The approach is based on the observation that in the long term the dynamics of a periodically driven quantum system can be represented in terms of a stationary effective Hamiltonian. The effective dynamics can thus be endowed with novel driving-induced properties that were not originally present in the static (undriven) system. In particular, topologically nontrivial band structures can be realized. In this talk, I will present a scheme based on the inverse-driving-frequency expansion that allows to systematically calculate successive contributions to the effective Hamiltonian. In particular, second order terms describe artificial fluxes and topological band structures. These terms can be interpreted as processes where a particle tunnels twice during one driving period. The possibility to realize the fractional Chern insulating (FCI) phases in the presence of strong atomic interactions will further be discussed. The interplay of tunneling processes with particle interactions gives rise to new interaction terms of several distinct types. For bosonic atoms with onsite interactions, such additional interaction terms include nearest-neighbor density-density interactions introduced at the cost of weakened on-site repulsion as well as density-assisted tunneling. We numerically investigated the impact of the individual induced interaction terms on the stability of a bosonic FCI state at half filling of the lowest band and mapped out a phase diagram indicating the regimes where stabilization of fractional phases is most promising.

The highly excited Rydberg-state atoms, which can induce strong dipole-dipole interaction, are one of the good candidates for achieving single-photon-level optical manipulation. Electromagnetically induced transparency (EIT) is used to map the interaction onto an optical transition. We will present our systematic study of a cascade-type Rydberg EIT transition, formed by a ground state, an intermediate state, and a Rydberg state in a laser-cooled ⁸⁷Rb atomic ensemble. The transition from the ground to intermediate state is driven by a 780 nm laser field (probe), and that from the intermediate to Rydberg state is driven by 480 nm laser field (coupling). Our study addressed the effect of laser frequency fluctuation on the observed EIT spectra.

We consider a quantum system periodically driven with a strength which varies slowly on the scale of the driving period. The analysis is based on a general formulation of the Floquet theory relying on the extended Hilbert space. It is shown that the dynamics of the system can be described in terms of a slowly varying effective Floquet Hamiltonian that captures the long-term evolution, as well as rapidly oscillating micromotion operators. We obtain a systematic high-frequency expansion of all these operators. Generalizing the previous studies, the expanded effective Hamiltonian is now time-dependent and contains extra terms appearing due to changes in the periodic driving. The same applies to the micromotion operators which exhibit a slow temporal dependence in addition to the rapid oscillations. As an illustration, we consider a quantum-mechanical spin in an oscillating magnetic field with a slowly changing direction. The effective evolution of the spin is then associated with non-Abelian geometric phases reflecting the geometry of the extended Floquet space. The developed formalism is general and also applies to other periodically driven systems, such as shaken optical lattices with a time-dependent shaking strength, a situation relevant to the cold atom experiments. My presentation will be related to our recent publication [1].

1. V. Novičenko, E. Anisimovas, and G. Juzeliūnas, Floquet analysis of a quantum system with modulated periodic driving Phys. Rev. A 95, 023615 (2017).

The discovery of fractional quantum Hall effect (FQHE) in 2D electron gas gave rise to immense interest in topological phases of matter [1]. One of the most intriguing features of FQHE state is fractionally charged excitations which embody anyonic statistics. Even though the FQHE was first observed in GaAs-GaAlAs heterojunctions, experiments in optical lattices [2] allow much more controllable study of many-body systems, therefore allowing regimes that are impossible to realise in semiconductor based experiments.

Historically, FQHE comes from condensed matter systems, which can be characterized by a very large number of particles, as a consequence, theoretical studies were focused only on infinite or periodical Hamiltonians. However, few of the unanswered questions remain: can FQHE states be realised in minuscule lattices, containing only several sites in diameter, and what additional effects would open boundaries produce? In this work we try to tackle both of these questions.

Using numerical diagonalization of Harper-Hofstadter model we were able to observe localisation of fractional charge in a 9×6 square lattice with artificial magnetic flux. Unfortunately, close proximity to the lattice edges does not allow direct confirmation of fractional statistics.

References

1. E. J. Bergholtz, Z. Liu, Topological flat band models and fractional Chern insulators, International Journal of Modern Physics B 27(24), 1330017 (2013).
2. I. Bloch, J. Dalibard, W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).