Entanglement signature of the superradiant quantum phase transition
Arthur Vesperini1,2, Matteo Cini3, Roberto Franzosi1,2()
. DSFTA, University of Siena, Via Roma 56, 53100 Siena, Italy . INFN Sezione di Perugia, I-06123 Perugia, Italy . Dipartimento di Fisica Università di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy
Entanglement and quantum correlations between atoms are not usually considered key ingredients of the superradiant phase transition. Here we consider the Tavis−Cummings model, a solvable system of two-levels atoms, coupled with a single-mode quantized electromagnetic field. This system undergoes a superradiant phase transition, even in a finite-size framework, accompanied by a spontaneous symmetry breaking, and an infinite sequence of energy level crossings. We find approximated expressions for the ground state, its energy, and the position of the level crossings, valid in the limit of a very large number of photons with respect to that of the atoms. In that same limit, we find that the number of photons scales quadratically with the coupling strength, and linearly with the system size, providing a new insight into the superradiance phenomenon. Resorting to novel multipartite measures, we then demonstrate that this quantum phase transition is accompanied by a crossover in the quantum correlations and entanglement between the atoms (qubits). The latters therefore represent suited order parameters for this transition. Finally, we show that these properties of the quantum phase transition persist in the thermodynamic limit.
Fig.1 Minima energy eigenvalues versus , for the first fifty multiplets. The figure refers to a system of qubits and with .
Fig.2 The figure shows a zoom the energy levels (continuous line), (dashed line), (dotted line), (dot-dash line), (gray dot-dash line) versus . The parameters are the same as for Fig.1.
Fig.3 The figure compares the plots of the energy level versus , corresponding to the minimum energy eigenvalue of the eigenspace , . Here we have considered a system with qubits and with . In continuous black line, we report derived by numeric diagonalization of the full Hamiltonian (7) and in dashed red line the approximated level given in Eq. (13). The agreement between the two curves is very good.
Fig.4 The figure reports the excitation number of the ground state, as a function of the coupling , derived by numerical calculations, in the cases . Line shows the asymptotic prediction (18), valid in the limit .
Fig.5 The figure reports the excitation number per qubit of the ground state, as a function of the coupling , derived by numerical calculations. All the curves, corresponding to the cases , are completely overlapping, and represented by the single red curve. The dotted line shows the asymptotic prediction drawn from Eq. (18), valid in the limit .
Fig.6 The figure plots the QCD per qubits versus , derived by numerical calculations. The lines refer to the cases of a system with qubits. Line shows the asymptotic prediction (31), valid in the limit .
Fig.7 The figure reports the purity as a function of the coupling , derived by numerical calculations, in the cases .
Fig.8 The figure reports the rescaled QCD per qubit as a function of the coupling , derived by numerical calculations, in the cases . Line shows the asymptotic prediction (31), valid in the limit .
Fig.9 The figure plots the total two-tangle per qubit versus , derived by numerical calculations, in the cases . The asymptotic prediction (31), valid in the limit , yields uniformly null total two tangle, for all values of .
1
Arute F., Arya K., Babbush R., Bacon D., C. Bardin J., et al.. Quantum supremacy using a programmable superconducting processor, Nature 574(7779), 505 (2019) https://doi.org/10.1038/s41586-019-1666-5
2
P. Dowling J. and J. Milburn G., Quantum technology: The second quantum revolution, Philos. T. R. Soc. A 361(1809), 1655 (2003) https://doi.org/10.1098/rsta.2003.1227
Y. Gyhm J., Šafránek D., and Rosa D., Quantum charging advantage cannot be extensive without global operations, Phys. Rev. Lett. 128(14), 140501 (2022) https://doi.org/10.1103/PhysRevLett.128.140501
5
Alicki R. and Fannes M., Entanglement boost for extractable work from ensembles of quantum batteries, Phys. Rev. E 87(4), 042123 (2013) https://doi.org/10.1103/PhysRevE.87.042123
6
C. Binder F., Vinjanampathy S., Modi K., and Goold J., Quantacell: Powerful charging of quantum batteries, New J. Phys. 17(7), 075015 (2015) https://doi.org/10.1088/1367-2630/17/7/075015
7
Campaioli F., A. Pollock F., C. Binder F., Céleri L., Goold J., Vinjanampathy S., and Modi K., Enhancing the charging power of quantum batteries, Phys. Rev. Lett. 118(15), 150601 (2017) https://doi.org/10.1103/PhysRevLett.118.150601
8
Y. Gyhm J., Šafránek D., and Rosa D., Quantum charging advantage cannot be extensive without global operations, Phys. Rev. Lett. 128(14), 140501 (2022) https://doi.org/10.1103/PhysRevLett.128.140501
9
J. Hwang M., Puebla R., and B. Plenio M., Quantum phase transition and universal dynamics in the Rabi model, Phys. Rev. Lett. 115(18), 180404 (2015) https://doi.org/10.1103/PhysRevLett.115.180404
T. Jaynes E. and W. Cummings F., Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proc. IEEE 51(1), 89 (1963) https://doi.org/10.1109/PROC.1963.1664
13
Retzker A., Solano E., and Reznik B., Tavis−Cummings model and collective multiqubit entanglement in trapped ions, Phys. Rev. A 75(2), 022312 (2007) https://doi.org/10.1103/PhysRevA.75.022312
14
Feng M., P. Zhong Y., Liu T., L. Yan L., L. Yang W., Twamley J., and Wang H., Exploring the quantum critical behaviour in a driven Tavis–Cummings circuit, Nat. Commun. 6(1), 7111 (2015) https://doi.org/10.1038/ncomms8111
Ghoshal A., Das S., Sen(De) A., and Sen U., Population inversion and entanglement in single and double glassy Jaynes−Cummings models, Phys. Rev. A 101(5), 053805 (2020) https://doi.org/10.1103/PhysRevA.101.053805
Fischer K., Sun S., Lukin D., Kelaita Y., Trivedi R., and Vuckovic J., Pulsed coherent drive in the Jaynes−Cummings model, Phys. Rev. A 98(2), 021802 (2018) https://doi.org/10.1103/PhysRevA.98.021802
19
J. Hwang M. and B. Plenio M., Quantum phase transition in the finite Jaynes−Cummings lattice systems, Phys. Rev. Lett. 117(12), 123602 (2016) https://doi.org/10.1103/PhysRevLett.117.123602
20
Bužek V., Orszag M., and Roško M., Instability and entanglement of the ground state of the Dicke model, Phys. Rev. Lett. 94(16), 163601 (2005) https://doi.org/10.1103/PhysRevLett.94.163601
21
Castaños O., López-Peña R., Nahmad-Achar E., G. Hirsch J., López-Moreno E., and E. Vitela J., Coherent state description of the ground state in the Tavis–Cummings model and its quantum phase transitions, Phys. Scr. 79(6), 065405 (2009) https://doi.org/10.1088/0031-8949/79/06/065405
22
Dong C.Zhang Y., Entanglement of atoms in Tavis−Cummings model, J. Shanghai University 10(3), 215 (2006) (English Edition)
23
E. Tessier T., H. Deutsch I., Delgado A., and Fuentes-Guridi I., Entanglement sharing in the two-atom Tavis−Cummings model, Phys. Rev. A 68(6), 062316 (2003) https://doi.org/10.1103/PhysRevA.68.062316
24
Youssef M., Metwally N., and S. F. Obada A., Some entanglement features of a three-atom Tavis–Cummings model: A cooperative case, J. Phys. At. Mol. Opt. Phys. 43(9), 095501 (2010) https://doi.org/10.1088/0953-4075/43/9/095501
25
Vesperini A., Bel-Hadj-Aissa G., and Franzosi R., Entanglement and quantum correlation measures for quantum multipartite mixed states, Sci. Rep. 13(1), 2852 (2023) https://doi.org/10.1038/s41598-023-29438-7
26
Roopini V. and Radhakrishnan R., Implementation of Tavis−Cummings model in solid-state defect qubits: Diamond nitrogenvacancy center, Mater. Today Proc. 27, 446 (2020) https://doi.org/10.1016/j.matpr.2019.11.266
Cocchiarella D., Scali S., Ribisi S., Nardi B., Bel-Hadj-Aissa G., and Franzosi R., Entanglement distance for arbitrary m-qudit hybrid systems, Phys. Rev. A 101(4), 042129 (2020) https://doi.org/10.1103/PhysRevA.101.042129
29
Nourmandipour A., Vafafard A., Mortezapour A., and Franzosi R., Entanglement protection of classically driven qubits in a lossy cavity, Sci. Rep. 11(1), 16259 (2021) https://doi.org/10.1038/s41598-021-95623-1
30
Vafafard A., Nourmandipour A., and Franzosi R., Multipartite stationary entanglement generation in the presence of dipole−dipole interaction in an optical cavity, Phys. Rev. A 105(5), 052439 (2022) https://doi.org/10.1103/PhysRevA.105.052439
31
Arthur V., Geometry, Topology, and Dynamics of Many-Body Systems: Quantum and Classical Perspectives, PhD thesis, U. Siena, Aix-Marseille University, 2023
Bel-Hadj-Aissa G., Geometric study of classical phase transitions and quantum entanglement, PhD thesis, U. Siena, 2023
34
Vesperini A., Bel-Hadj-Aissa G., Capra L., and Franzosi R., Unveiling the geometric meaning of quantum entanglement: Discrete and continuous variable systems, Front. Phys. 19(5), 51204 (2024) https://doi.org/10.1007/s11467-024-1403-x
Tavis M. and W. Cummings F., Exact solution for an n-molecule — radiation-field hamiltonian, Phys. Rev. 170(2), 379 (1968) https://doi.org/10.1103/PhysRev.170.379
40
D. Leroux I., H. Schleier-Smith M., and Vuletic V., Implementation of cavity squeezing of a collective atomic spin, Phys. Rev. Lett. 104(7), 073602 (2010) https://doi.org/10.1103/PhysRevLett.104.073602
Bužek V., Orszag M., and Roško M., Instability and entanglement of the ground state of the Dicke model, Phys. Rev. Lett. 94(16), 163601 (2005) https://doi.org/10.1103/PhysRevLett.94.163601
Koashi M., Bužek V., and Imoto N., Entangled webs: Tight bound for symmetric sharing of entanglement, Phys. Rev. A 62(5), 050302 (2000) https://doi.org/10.1103/PhysRevA.62.050302
