Abstract
Superpositions of macroscopically distinct quantum states, introduced in Schrödinger's famous Gedankenexperiment, are an epitome of quantum ‘strangeness’ and a natural tool for determining the validity limits of quantum physics. The optical incarnation of Schrödinger's cat (SC)—the superposition of two oppositeamplitude coherent states—is also the backbone of continuousvariable quantum information processing. However, the existing preparation methods limit the amplitudes of the component coherent states, which curtails the state's usefulness for fundamental and practical applications. Here, we convert a pair of negative squeezed SC states of amplitude 1.15 to a single positive SC state of amplitude 1.85 with a success probability of ∼0.2. The protocol consists in bringing the initial states into interference on a beamsplitter and a subsequent heralding quadrature measurement in one of the output channels. Our technique can be realized iteratively, so arbitrarily high amplitudes can, in principle, be reached.
Main
In Schrödinger's proposal, the life and death of a cat are entangled with the state of a decaying atom, which results in a macroscopic quantumsuperposition state^{1}. This setting, originally used as a metaphor to demonstrate the absurdity of the newborn quantum theory in the macroscopic domain, remained a matter of thought experiment for about 50 years. As quantum physics matured, this paradox was revisited; nowadays, Schrödinger's cat (SC) is being emulated in diverse physical systems. It is expected to help answer a fundamental question^{2,3,4}: at what degree of macroscopicity, if any, does the world stop being quantum?
In optics, the SC state corresponds to a superposition of coherent states ±α〉, which are considered the most classical of all states of light^{5}. In such a superposition, the fields of the electromagnetic wave point in two opposite directions at the same time, which resembles the superposition of dead and alive states of a cat in the original concept. The size parameter in this case is the amplitude α. For the SC to be macroscopic, α has to be much larger than the quantum uncertainty of the position observable in the coherent state^{5,6,7}.
Aside from the fundamental interest, optical SC states are useful in applied quantum science. They can serve as a basis for quantum computation^{8,9}, metrology^{10,11}, teleportation and cryptography^{12,13,14}. Most of these applications require the involved coherent states to be of reasonably high amplitudes. For example, encoding a qubit in the coherent state basis ±α〉 is practical only when these states are nearly orthogonal, that is, when α ≥ 2 (ref. 8). Although a faulttolerant quantum computation scheme optimized at α ≈ 1.6 has been proposed, it requires a significant resource overhead^{9}.
The above motivation to build optical SC states inspired a significant experimental drive^{15,16,17,18,19,20,21,22,23,24,25,26,27}. Most of the existing experiments are based on photon subtraction from the squeezed vacuum state^{19,20,21,22,23,24,25} or on quantumstate engineering within the subspace of three lower Fock states^{15,16,17,18}. The state obtained by these methods approximates SCs reasonably well only for relatively small amplitudes. A better performance is offered by the method of preparing SCs from multiphoton Fock states^{26}, but these states are, themselves, difficult to prepare in a reliable and scalable fashion.
In this work, we implement an alternative for the directpreparation approach mentioned above. Our technique probabilistically converts a pair of SCs into a single SC state with an amplitude greater than that of the initial ones by a factor of . Our method can be applied in an iterative manner^{27,28}, which allows one to prepare an SC state of, in principle, any desirable amplitude—given that sufficiently many initial SCs are available at the inception stage.
The idea of the method was proposed by Lund et al.^{29} and further developed by Laghaout et al.^{28}. Let the initial SC state be a superposition of coherent states of real amplitude α be:
where N is the normalization factor. Suppose a pair of identical, either positive or negative, states ( equation (1)) is put to interference on a symmetric beamsplitter. Let the relative phase of the inputs be such that equal coherent states interfere constructively in its output mode 1 and destructively in mode 2, whereas oppositeamplitude coherent states behave in the converse fashion. The twomode output state of the beam splitter is then^{29}:
where 0〉 is the vacuum state. If we now perform a measurement on mode 2 to distinguish the states 0〉 and , and detect the vacuum state, mode 1 will collapse onto the positive SC state of amplitude (ref. 29).
The required conditioning can be realized by homodyne measurement of the position quadrature in mode 2 and looking for the null result. Indeed, the wavefunction of state is a sum of two Gaussians of width centred, respectively, at , whereas the wavefunction of the vacuum state is the same Gaussian centred at X = 0. Therefore, for α ≫ 1, the probability of observing X = 0 is much higher in the vacuum state than in the state . A precise calculation yields:
For the initial SC amplitude α = 1, which is easily attained by present technology^{15,16,19,23,24,25,26,27}, state (3) has a fidelity of 99% with the ideal state^{28} (throughout this paper, the fidelity between the states with density operators ρ_{1} and ρ_{2} is defined as ).
In experimental practice, the SC amplification event is heralded by observing X within a finite nearzero interval. In our case of α = 1.15, we set the boundaries of this interval at 0.3, which gives a success probability of p ≈ 0.2 without significant additional fidelity loss. Remarkably, the acceptable width of this interval and, consequently, the success probability of this protocol increases with α, reaching p ≈ 1/2 for α ≥ 2. This is because the peaks in the wavefunction of the SC move further away from X = 0. Moreover, the fidelity of amplification also increases with α due to the diminishing weight of the vacuum component in equation (3).
The experimental setup for realizing the above protocol is shown in Fig. 1. The initial SCs in two spatially distinct light modes are generated by photon subtraction from squeezed vacuum states^{19,30}. As the squeezed vacuum is approximated by a positive SC (equation (1a)), the photon annihilation flips the sign in front of the −α〉 component, which converts the state to a negative squeezed SC (equation (1b)) of a larger amplitude^{19,20}. The obtained states are characterized by optical homodyne tomography^{31,32}. Corrected for the total quantum efficiency of 62% (Methods), they have a fidelity of 84% with an ideal state SC_{−}[1.15]〉 squeezed by 1.74 dB. The Wigner functions of the experimentally reconstructed states and the bestfit state are shown in Fig. 2 (right column). Throughout this paper, we follow the concept of Ourjoumtsev et al.^{26} by fitting our experimentally acquired states with squeezed SCs. The rationale behind this convention is that squeezing does not affect the macroscopicity of the superposition^{33} and can be undone by a local unitary operation.
Simultaneous preparation of the two initial SC states is heralded by coincident clicks of singlephoton counting modules (SPCMs) 1 and 2. The states are then mixed on a symmetric beamsplitter. The optical phase difference between them is stabilized actively (Methods) to make sure that the output state of the beamsplitter is described by equation (2). Subsequently, one of the beamsplitter output modes (mode 2) is subjected to homodyne measurement of the position quadrature. Conditioned on a nearzero measurement result (X ≤ 0.3) in mode 2, the state of mode 1 is subjected to quantum tomography by means of another homodyne detector. Out of a total of 40,000 SPCM coincidence events collected, 8,000 satisfied this condition, which corresponds to the protocol's success probability of p = 0.2.
The tomographic reconstruction result is shown in Fig. 2a (right). The amplified state, corrected for a 62% detection efficiency, has a fidelity of 77% with SC_{+}[1.85]〉 squeezed by 3.04 dB, displayed in Fig. 2b (right). The additional squeezing of the output SC compared with the input is a result of the nonideality of the homodyne detection in mode 2 as well as the imperfection of the initial SC.
The Wigner functions of the SC states have a characteristic shape that consists of two positive Gaussian peaks associated with the individual coherent state constituents and a highly nonclassical ‘interference fringe’ pattern between them. Our observations are consistent with this description. In the initial SC states, the Gaussian peaks are quite close to each other, so the Wigner function of the initial state resembles that of the squeezed single photon^{19}. For the amplified SC, the peaks are separated further, so one can more clearly distinguish them from the interference pattern in between, with the latter becoming more prominent. This effect on the Wigner function is also quite evident without the efficiency correction (Fig. 2a, left insets).
The protocol demonstrated here constitutes an instrument to convert a pair of SC states into a single largeramplitude positive SC state. The probability of success p is directly related to the width of the quadrature selection band in output mode 2 of the beamsplitter, and asymptotically increases to 1/2 for high amplitudes. As shown, the fidelity of the amplified SC does not significantly decrease with respect to that of the initial one, which permits the application of the protocol in an iterative fashion.
A single realization of our protocol produces optical SCs with amplitudes that are comparable to the highest ever achieved, including other physical systems, such as microwave^{34} and circuit^{35} cavity quantum electrodynamics settings. Iterating our protocol for n stages will further increase the SC amplitude by a factor of α′/α = 2^{n/2}. As each implementation of the protocol would require two input SCs, a total of 1 + 2 + … + 2^{n−1} = 2^{n} − 1 implementations are needed for n stages, with the corresponding success probability of ${p}^{{2}^{n1}}={p}^{({\alpha}^{\prime}/\alpha )21}$. This estimate can be increased by using wider acceptance intervals at advanced stages of the protocol with α ≥ 2. For example, amplifying the SC state from α = 1.4 to α′ = 4 would require, on average, ∼5,000 copies of the initial SC per one copy of the output.
The multistage version of the breeding protocol would function best with ondemand SC sources. The recently developed highefficiency, reproducible singlephoton sources^{36,37} are promising in this context. The photons obtained from these sources can be subjected to squeezing to generate lowamplitude SCs, or used directly as the input states of the first stage^{27}. Alternatively, one can use heralded SC sources, such as those employed in this experiment, combined with optical quantum memory in a setting similar to the quantum repeater^{38}. This will change the scaling of the overall success probability with respect to the target amplitude from exponential to polynomial.
It is instructive to compare our method with the techniques for the heralded preparation of arbitrary Fockstate superpositions by engineering the measurement in the idler channel of parametric downconversion^{17,18}. Theoretically, any quantum state, including SCs, can be decomposed into the Fock basis and thus prepared in this fashion. In practice, however, these methods operate in the limit of low probability to generate a photon pair. Hence, in contrast to the method developed here, they feature prohibitively low success probabilities for any states beyond the 2–3 photon subspace.
Methods
Degenerate parametric downconversion takes place in periodically poled potassium titanyl phosphate crystals (PPKTP, Raicol) under typeI phasematching conditions. Each of the two crystals is pumped with ∼25 mW frequencydoubled radiation of the master laser (Ti:Sapphire Coherent Mira 900D, with a wavelength of 780 nm, repetition rate of 76 MHz and pulse width of ∼1.5 ps). In each nonlinear crystal, a 1.7 dB singlemode squeezed vacuum state is generated^{39}.
For the preparation of the initial SCs, 10% of the energy from these squeezed vacuum states is ‘tapped’ by beamsplitters and directed to SPCMs (Excellitas) via fibre interfaces. The preparation rate of each SC state is ∼10 kHz, which results in a ∼2 Hz rate of coincidence events.
The setup requires two phaselock loops (PLLs), one to keep the input SCs in phase with each other and the other to ensure that the homodyne detector in mode 2 measures the X quadrature. To generate the feedback signal, both PLLs use the data from the homodyne detectors without conditioning on the SPCM events (which we refer to as ‘nontriggered’). The first PLL is set to minimize the Einstein–Podolsky–Rosentype quadrature correlations, which show up in the detectors' measurements when the phases are misaligned^{39}. The feedback signal is applied to a piezoelectric transducer in one of the initial state's paths (PZT_{1} in Fig. 1).
If the first PLL functions properly, the nontriggered output of the beamsplitter constitutes two unentangled singlemode momentumsqueezed states. The second PLL can therefore be set to keep the variance of the mode 2 quadrature measurements at the maximum. The feedback signal is applied to the corresponding local oscillator phase via PZT_{2} (Fig. 1).
The reconstruction of both the initial and amplified SC states is performed using the iterative maximumlikelihood algorithm^{31,32}. The local oscillator phase is varied by PZT_{3} and its timedependent value is extracted from the variance of the nontriggered quadrature data, which corresponds to the singlemode squeezed state. To perform the tomography of the initial SCs, the reflectivity of the central beamsplitter is set to zero.
The total quantum efficiency of state detection, 62%, is determined from the analysis of these SC states^{40}. The main efficiency reduction factors are optical losses (10%, not counting the tapping beamsplitter), mode matching between the signal and local oscillator (81%) and the efficiency of the homodyne detector^{41} (86%).
The observed experimental imperfections can be explained by the model of Ourjoumtsev et al.^{19}. According to this model, the SPCMs' dark events, as well as imperfect matching between the modes detected by the SPCMs and the squeezed mode, results in an imperfect photon subtraction. The density matrix of the input SC is thus given by , where ρ_{sq} is the density matrix of the squeezed vacuum state subjected to 10% of losses on the tapping beamsplitter. The best fidelity of 98% between the model and the experimentally reconstructed state was found for ξ = 0.9.
Subsequently, the ‘breeding’ protocol was applied to the modelled input SC, taking into account the efficiency of the heralding homodyne detector. The best fit gives a 97% fidelity with the experimentally obtained density matrix.
If the parameter ξ was equal to 0.99 and the tapping beamsplitter reflectivity was 0.05 rather than 0.1, which can be achieved by modern experimental methods, the fidelity between the ideal cat states and the output (input) states calculated using the same model would be 95% (96%). In other words, the fidelity loss observed for the amplified SC in the present experiment mainly results from the imperfections in the initial SC rather than the ‘breeding’ operation itself.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author on reasonable request.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
 1
Schrödinger, E. Die gegenwärtige situation in der quantenmechanik. Naturwissenschaften 23, 807–812 (1935).
 2
Haroche, S. Nobel lecture: controlling photons in a box and exploring the quantum to classical boundary. Rev. Mod. Phys. 85, 1083–1102 (2013).
 3
Wineland, D. J. Nobel lecture: superposition, entanglement, and raising Schrödinger's cat. Rev. Mod. Phys. 85, 1103–1114 (2013).
 4
Markus, A. & Hornberger, K. Testing the limits of quantum mechanical superpositions. Nat. Phys. 10, 271–277 (2014).
 5
Leonhardt, U. Measuring Quantum States of Light (Cambridge Univ. Press, 1997).
 6
Leggett, A. J. Testing the limits of quantum mechanics: motivation, state of play, prospects. J. Phys. Condens. Matter 14, R415–R451 (2002).
 7
Lvovsky, A. I., Ghobadi, R., Chandra, A., Prasad, A. S. & Simon, C. Observation of micro–macro entanglement of light. Nat. Phys. 9, 541–544 (2013).
 8
Ralph, T. C., Gilchrist, A., Milburn, G. J., Munro, W. J. & Glancy, S. Quantum computation with optical coherent states. Phys. Rev. A 68, 042319 (2003).
 9
Lund, A. P., Ralph, T. C. & Haselgrove, H. L. Faulttolerant linear optical quantum computing with smallamplitude coherent states. Phys. Rev. Lett. 100, 030503 (2007).
 10
Joo, J., Munro, W. J. & Spiller, T. P. Quantum metrology with entangled coherent states. Phys. Rev. Lett. 107, 083601 (2011).
 11
Facon, A. et al. A sensitive electrometer based on a Rydberg atom in a Schrödingercat state. Nature 535, 262–265 (2016).
 12
Lee, S.W. & Jeong, H. Neardeterministic quantum teleportation and resourceefficient quantum computation using linear optics and hybrid qubits. Phys. Rev. Lett. 87, 022326 (2012).
 13
Sangouard, N. et al. Quantum repeaters with entangled coherent states. J. Opt. Soc. Am. B 27, A137–A145 (2010).
 14
Brask, J. B., Rigas, I., Polzik, E. S., Andersen, U. L. & Sørensen, A. S. Hybrid longdistance entanglement distribution protocol. Phys. Rev. Lett. 105, 160501 (2010).
 15
Huang, K. et al. Optical synthesis of largeamplitude squeezed coherentstate superpositions with minimal resource. Phys. Rev. Lett. 115, 023602 (2015).
 16
Ulanov, A. E., Fedorov, I. A., Sychev, D., Grangier, P. & Lvovsky, A. I. Losstolerant state engineering for quantumenhanced metrology via the reverse Hong–Ou–Mandel effect. Nat. Commun. 7, 11925 (2016).
 17
Bimbard, E., Jain, N., MacRae, A. & Lvovsky, A. I. Quantumoptical state engineering up to the twophoton level. Nat. Photon. 4, 243–247 (2010).
 18
Yukawa, M. et al. Generating superposition of upto three photons for continuous variable quantum information processing. Opt. Express 21, 5529–5535 (2013).
 19
Ourjoumtsev, A., TualleBrouri, R., Laurat, J. & Grangier, P. Generating optical Schrödinger kittens for quantum information processing. Science 312, 83–86 (2006).
 20
NeergaardNielsen, J. S., Melholt Nielsen, B., Hettich, C., Mølmer, K. & Polzik, E. S. Generation of a superposition of odd photon number states for quantum information networks. Phys. Rev. Lett. 97, 083604 (2006).
 21
Wakui, K., Takahashi, H., Furusawa, A. & Sasaki, M. Photon subtracted squeezed states generated with periodically poled KTiOPO4 . Opt. Express 15, 3568–3574 (2007).
 22
Ourjoumtsev, A., Ferreyrol, F., TualleBrouri, R. & Grangier, P. Preparation of nonlocal superpositions of quasiclassical light states. Nat. Phys. 5, 189–192 (2009).
 23
Gerrits, T. et al. Generation of optical coherent state superpositions by numberresolved photon subtraction from squeezed vacuum. Phys. Rev. A 82, 031802(R) (2010).
 24
Takahashi, H. et al. Generation of largeamplitude coherentstate superposition via ancillaassisted photon subtraction. Phys. Rev. Lett. 101, 233605 (2008).
 25
Dong, R. et al. Generation of picosecond pulsed coherent state superpositions. J. Opt. Soc. Am. B 31, 1192–1201 (2014).
 26
Ourjoumtsev, A., Jeong, H., TualleBrouri, R. & Grangier, P. Generation of optical Schrödinger cats from photon number states. Nature 448, 1784–1786 (2007).
 27
Etesse, J., Bouillard, M., Kanseri, B. & TualleBrouri, R. Experimental generation of squeezed cat states with an operation allowing iterative growth. Phys. Rev. Lett. 114, 193602 (2015).
 28
Laghaout, A. et al. Amplification of realistic Schrödingercatstatelike states by homodyne heralding. Phys. Rev. A 87, 043826 (2013).
 29
Lund, A. P., Jeong, H., Ralph, T. C. & Kim, M. S. Conditional production of superpositions of coherent states with inefficient photon detection. Phys. Rev. A 70, 020101 (2004).
 30
Dakna, M., Anhut, T., Opatrny, T., Knöll, L. & Welsch, D.G. Generating Schrödingercatlike states by means of conditional measurements on a beam splitter. Phys. Rev. A 55, 3184–3194 (1997).
 31
Lvovsky, A. I. & Raymer, M. G. Continuousvariable optical quantumstate tomography. Rev. Mod. Phys. 81, 299–332 (2009).
 32
Lvovsky, A. I. Iterative maximumlikelihood reconstruction in quantum homodyne tomography. J. Opt. B 6, S556–S559 (2004).
 33
Lee, C. W. & Jeong, H. Quantification of macroscopic quantum superpositions within phase space. Phys. Rev. Lett. 106, 220401 (2011).
 34
Deléglis, S. et al. Reconstruction of nonclassical cavity field states with snapshots of their decoherence. Nature 455, 510–515 (2008).
 35
Vlastakis, B. et al. Deterministically encoding quantum information using 100photon Schrödinger cat states. Science 342, 607–610 (2013).
 36
Ding, X. et al. Ondemand single photons with high extraction efficiency and nearunity indistinguishability from a resonantly driven quantum dot in a micropillar. Phys. Rev. Lett. 116, 020401 (2016).
 37
Somaschi, N. et al. Nearoptimal singlephoton sources in the solid state. Nat. Photon. 10, 340–345 (2016).
 38
Lvovsky, A. I., Sanders, B. C. & Tittel, W. Optical quantum memory. Nat. Photon. 3, 706–714 (2009).
 39
Fedorov, I. A., Ulanov, A. E., Kurochkin, Y. & Lvovsky, A. I. Synthesis of the Einstein–Podolsky–Rosen entanglement in a sequence of two singlemode squeezers. Opt. Lett. 42, 132–134 (2017).
 40
Berry, D. W. & Lvovsky, A. I. Linearoptical processing cannot increase photon efficiency. Phys. Rev. Lett. 105, 203601 (2010).
 41
Kumar, R. et al. Versatile wideband balanced detector for quantum optical homodyne tomography. Opt. Commun. 285, 5259–5267 (2012).
Acknowledgements
We thank Y. Kurochkin and A. Turlapov for discussions. We acknowledge financial support from the Ministry of Education and Science of the Russian Federation (Agreement 14.582.21.0009, ID RFMEFI58215X0009). A.I.L. is supported by the Natural Sciences and Engineering Research Council of Canada and is a Canadian Institute for Advanced Research Fellow.
Author information
Affiliations
Contributions
All the authors participated in the conception and planning of the project, theoretical analysis and writing of the paper. The experiment was performed by D.V.S., A.E.U., A.A.P., I.A.F. and M.W.R. The data were analysed by D.V.S., A.E.U., I.A.F. and A.I.L.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
About this article
Cite this article
Sychev, D., Ulanov, A., Pushkina, A. et al. Enlargement of optical Schrödinger's cat states. Nature Photon 11, 379–382 (2017). https://doi.org/10.1038/nphoton.2017.57
Received:
Accepted:
Published:
Issue Date:
Further reading

Feasible and economical scheme to entangle a polarized coherent state and a polarized photon
Optik (2021)

Heralded Interaction Control between Quantum Systems
Physical Review Letters (2020)

Quantum Mechanics and Its Interpretations: A Defense of the Quantum Principles
Foundations of Physics (2020)

Quasiprobability distributions for quantumoptical coherence and beyond
Physica Scripta (2020)

Teleportation of a multiphoton qubit using hybrid entanglement with a losstolerant carrier qubit
Physical Review A (2020)