Abstract
Recently, using conditioning approaches on the high-harmonic generation process induced by intense laser-atom interactions, we have developed a new method for the generation of optical Schrödinger cat states (Lewenstein et al. in Nat Phys, 17 1104–1108, 2021. https://doi/10.1038/s41567-021-01317-w). These quantum optical states have been proven to be very manageable as, by modifying the conditions under which harmonics are generated, one can interplay between kitten and genuine cat states. Here, we demonstrate that this method can also be used for the development of new schemes towards the creation of optical Schrödinger cat states, consisting of the superposition of three distinct coherent states. Apart from the interest these kind of states have on their own, we additionally propose a scheme for using them towards the generation of large cat states involving the sum of two different coherent states. The quantum properties of the obtained superpositions aim to significantly increase the applicability of optical Schrödinger cat states for quantum technology and quantum information processing.
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The code used for the generation of the presented figures is made available in [38] under the Creative Commons Attribution-ShareAlike 4.0 (CC BY-SA 4.0) license.
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Acknowledgements
ICFO group acknowledges support from ERC AdG NOQIA, State Research Agency AEI (“Severo Ochoa” Center of Excellence CEX2019-000910-S, Plan National FIDEUA PID2019-106901GB-I00/10.13039/501100011033, FPI, QUANTERA MAQS PCI2019-111828-2/10.13039/501100011033), RETOS-QUSPIN, Fundació Privada Cellex, Fundació Mir-Puig, Generalitat de Catalunya (AGAUR Grant No. 2017 SGR 1341, CERCA program, QuantumCAT\U16-011424, co-funded by ERDF Operational Program of Catalonia 2014-2020), EU Horizon 2020 FET-OPEN OPTOLogic (Grant No 899794), and the National Science Centre, Poland (Symfonia Grant No. 2016/20/W/ST4/00314), Marie Sk\l odowska-Curie grant STREDCH No 101029393, “La Caixa” Junior Leaders fellowships (ID100010434), and EU Horizon 2020 under Marie Sk\l odowska-Curie grant agreement No 847648 (LCF/BQ/PI19/11690013, LCF/BQ/PI20/11760031, LCF/BQ/PR20/11770012). J.R-D. acknowledges support from the Secretaria d’Universitats i Recerca del Departament d’Empresa i Coneixement de la Generalitat de Catalunya, as well as the European Social Fund (L’FSE inverteix en el teu futur)–FEDER. P.T. group acknowledges LASERLABEUROPE (H2020-EU.1.4.1.2 Grand ID 654148)), FORTH Synergy Grant AgiIDA (Grand No.: 00133). ELI-ALPS is supported by the European Union and co-financed by the European Regional Development Fund (GINOP Grant No. 2.3.6-15-2015-00001).
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Appendix: Probability of success in the generalization of the conditioning approach
Appendix: Probability of success in the generalization of the conditioning approach
Here, we present a more detailed analysis about the probability of success plotted in Figs. 7 and 8. In the text, we are interested in measuring even or odd number of photons in one of the output modes of the beam splitter, as this operation projects the other mode into a cat state, which differ in a relative phase of \(\pi\) depending on the specific measurement that is performed. Thus, the set of positive operator-valued measurements characterizing the operations that we can implement is
which project the second mode into the vacuum state, an even number of photons and an odd number of photons, respectively.
Each of the measurements described in Eq. (32) have a probability associated, so we define the probability of success \(\mathcal {P}\) as the probability of successfully performing one of such measurements. For instance, the probability of successfully measuring an even number of photons is given by
where \(\rho\) is the density matrix characterizing the state of our system, and \(\mathcal {P}_\text {total}\) is given as the sum of all the possible measurements that we can perform, i.e.
With all this set, let us consider the architecture presented in the main text used for the generation of enlarged cat states. Using a beam splitter characterized by the mixing angle \(\theta\) and a coherent state of the form \(\left| (\alpha - \delta \alpha ) \tan (\theta ) \right\rangle\), the final state after the beam splitter is given by
where \({\tilde{\alpha }} = (\alpha -\delta \alpha )/\cos \theta\). Thus, the measurement of an even number of photons projects the state in Eq. (35) into
the measurement of an odd number of photons projects the state into
and, finally, the measurement of zero number of photons leads to
Thus, the probability of success for each of the three measurements that we can do is given by
and
Note that the case of \(\delta \alpha = 0\) leads to a divergence in the obtained equations. In that case, the applied measurement does not make sense at all because the case of \(\delta \alpha = 0\) corresponds to the situation in which we do not have HHG processes taking place. Thus, no optical Schrödinger cat state is being generated.
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Rivera-Dean, J., Stammer, P., Pisanty, E. et al. New schemes for creating large optical Schrödinger cat states using strong laser fields. J Comput Electron 20, 2111–2123 (2021). https://doi.org/10.1007/s10825-021-01789-2
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DOI: https://doi.org/10.1007/s10825-021-01789-2