Kjetil Børkje

• Ryen, Kai & Børkje, Kjetil (2026). Witnesses of genuine multipartite entanglement and nonlocal measurement backaction for Raman-scattering quantum systems. Physical Review A (PRA). ISSN 2469-9926. 113(3). doi: 10.1103/5wv4-fb72. Full text in Research Archive Show summary

  Entanglement between remote quantum mechanical systems enables a range of quantum information tasks in communication, computation, and distributed sensing. Large numbers of entangled subsystems also require experimentally accessible and practically feasible methods of verifying the genuine, i.e., simultaneous, entanglement of all subsystems. We have derived a class of entanglement witnesses suitable for W states, which are states where a single excitation is coherently distributed across subsystems initially in their ground state or a state with low thermal occupation, e.g., via detection of a Raman-scattered photon. The entanglement is witnessed through violation of an inequality involving number statistics, which can be measured via detection of subsequent Raman-scattered photons. Unlike conventional, partially tomographic, witnesses, our method is experimentally accessible for both multipartite and continuous variable systems. The thermal robustness of the method is quantified by the initial thermal occupations for which violation occurs. As an alternative approach, we have derived an inequality which tests the nonlocal, or quantum coherent, nature of the photon measurement backaction which produces the W state. Violation of this alternative inequality implies the entanglement of the resulting state given the assumption of a separable initial state, under less stringent thermal constraints than the general entanglement witness. Our results are applicable to all Raman-scattering systems which can exhibit sufficient degrees of quantum indistinguishability.

• Ryen, Kai & Børkje, Kjetil (2024). Proposal for observing nonclassicality in highly excited mechanical oscillators by single photon detection. Physical Review A (PRA). ISSN 2469-9926. 109(4). doi: 10.1103/PhysRevA.109.043505. Full text in Research Archive
• Wang, Cheng; Banniard, Louise; Børkje, Kjetil; Massel, Francesco Pietro; Lepinay, Laure Mercier de & Sillanpää, Mika A. (2024). Ground-state cooling of a mechanical oscillator by a noisy environment. Nature Communications. 15(1). doi: 10.1038/s41467-024-51645-7. Full text in Research Archive Show summary

  Dissipation and the accompanying fluctuations are often seen as detrimental for quantum systems since they are associated with fast relaxation and loss of phase coherence. However, it has been proposed that a pure state can be prepared if external noise induces suitable downwards transitions, while exciting transitions are blocked. We demonstrate such a refrigeration mechanism in a cavity optomechanical system, where we prepare a mechanical oscillator in its ground state by injecting strong electromagnetic noise at frequencies around the red mechanical sideband of the cavity. The optimum cooling is reached with a noise bandwidth smaller than but on the order of the cavity decay rate. At higher bandwidths, cooling is less efficient as suitable transitions are not effectively activated. In the opposite regime where the noise bandwidth becomes comparable to the mechanical damping rate, damping follows the noise amplitude adiabatically, and the cooling is also suppressed.

• Banniard, Louise; Wang, Cheng; Stirpe, Davide; Børkje, Kjetil; Massel, Francesco Pietro & Lépinay, Laure Mercier de [Show all 7 contributors for this article] (2024). Optomechanics Driven by Noisy and Narrowband Fields. Journal of Low Temperature Physics. ISSN 0022-2291. 217, p. 720–734. doi: 10.1007/s10909-024-03220-z. Full text in Research Archive
• Børkje, Kjetil & Marin, F (2023). Quantum state purity versus average phonon number for characterization of mechanical oscillators in cavity optomechanics. Physical Review A (PRA). ISSN 2469-9926. 107(1). doi: 10.1103/PhysRevA.107.013502. Full text in Research Archive
• Ranfagni, A.; Børkje, Kjetil; Marino, F. & Marin, F (2022). Two-dimensional quantum motion of a levitated nanosphere. Physical Review Research (PRResearch). 4(3). doi: 10.1103/PhysRevResearch.4.033051. Full text in Research Archive Show summary

  We report on the two-dimensional (2D) dynamics of a levitated nanoparticle in an optical cavity. The motion of the nanosphere is strongly coupled to the cavity field by coherent scattering and heavily cooled in the plane orthogonal to the tweezer axis. Due to the characteristics of the 2D motion and the strong optomechanical coupling, the motional sideband asymmetry that reveals the quantum nature of the dynamics is not limited to mere scale factors between Stokes and anti-Stokes peaks, as customary in quantum optomechanics, but assumes a peculiar spectral dependence. We introduce and discuss an effective thermal occupancy that quantifies how close the system is to a minimum uncertainty state and allows us to consistently characterize the particle motion. By rotating the polarization angle of the tweezer beam we tune the system from a 1D cooling regime, where we achieve a best thermal occupancy of 0.51 ± 0.05, to a regime in which the fully 2D dynamics of the particle exhibit strong nonclassical properties. We achieve a strong 2D confinement with thermal occupancy of 3.4 ± 0.4 along the warmest direction and around one in the orthogonal one. These results represent a major improvement over previous experiments considering both the 1D and the 2D motion and pave the way toward the preparation of tripartite optomechanical entangled states and novel applications to directional force and displacement quantum sensing.

• Børkje, Kjetil; Massel, Francesco Pietro & Harris, Jack (2021). Nonclassical photon statistics in two-tone continuously driven optomechanics. Physical Review A (PRA). ISSN 2469-9926. 104(6). doi: 10.1103/PhysRevA.104.063507. Full text in Research Archive Show summary

  In cavity optomechanics, single-photon detection of Raman scattered photons can be a useful tool for observing nonclassical features of both radiation and motion. While this tool has been employed in experiments with pulsed laser driving of a cavity mode, it has not been readily accessible to experiments with continuous and constant laser driving. To address this, we present a study of a standard optomechanical system where the cavity mode is continuously driven at two different frequencies and where sideband photons are detected by single-photon detectors after frequency filtering the output from the cavity mode around its resonance frequency. We first derive the normalized second-order coherence associated with the detected photons and show that it contains signatures of the quantum nature of the mechanical mode which would be absent with only single-tone driving. To identify model-independent nonclassical features, we derive two inequalities for the sideband photon statistics that should be valid in any classical model of the system. We show that these inequalities are violated in the proposed setup. This is provided that the average phonon occupation number of the mechanical mode is sufficiently small, which in principle can be achieved through sideband cooling intrinsic to the setup. Violation of the first inequality means that there is no well-defined probability distribution of the Glauber-Sudarshan type for the cavity mode. In contrast, a violation of the second inequality means that there is no joint probability distribution for the cavity mode at two times separated by a finite interval, which originates from the noncommutativity of the motional quadratures of the mechanical mode. The proposed setup thus employs a mechanical oscillator in order to generate a steady-state source of nonclassical radiation.

• Børkje, Kjetil (2020). Critical quantum fluctuations and photon antibunching in optomechanical systems with large single-photon cooperativity. Physical Review A (PRA). ISSN 2469-9926. 101(5). doi: 10.1103/PhysRevA.101.053833. Full text in Research Archive Show summary

  A pertinent question in cavity optomechanics is whether reaching the regime of large single-photon cooperativity, where the single-photon coupling rate exceeds the geometric mean of the cavity and mechanical decay rates, can enable any new phenomena. We show that in some multimode optomechanical systems, the single-photon cooperativity can indeed be a figure of merit. We first study a system with one cavity mode and two mechanical oscillators, which combines the concepts of levitated optomechanics and coherent scattering with standard dispersive optomechanics. Later, we study a more complicated setup involving three cavity modes which does not rely on levitated optomechanics and only features dispersive optomechanical interactions with direct cavity driving. These systems can effectively realize the degenerate or the nondegenerate parametric oscillator models known from quantum optics, but in the unusual finite-size regime for the fundamental mode(s) when the single-photon cooperativity is large. We show that the response of these systems to a coherent optical probe can be highly nonlinear in probe power even for average photon occupation numbers below unity. The nonlinear optomechanical interaction has the peculiar consequence that the probe drive will effectively amplitude-squeeze itself. For large single-photon cooperativity, this occurs for small occupation numbers, which enables observation of nonclassical antibunching of the transmitted probe photons due to a destructive interference effect. Finally, we show that as the probe power is increased even further, the system enters a critical regime characterized by intrinsically nonlinear dynamics and non-Gaussian states.

• Børkje, Kjetil (2019). Theory of bifurcation amplifiers utilizing the nonlinear dynamical response of an optically damped mechanical oscillator. Physical Review A (PRA). ISSN 2469-9926. 99. doi: 10.1103/PhysRevA.99.023858. Full text in Research Archive Show summary

  We consider a standard optomechanical system where a mechanical oscillator is coupled to a cavity mode through the radiation pressure interaction. The oscillator is coherently driven at its resonance frequency, whereas the cavity mode is driven below its resonance, providing optical damping of the mechanical oscillations. We study the nonlinear coherent response of the mechanical oscillator in this setup. For large mechanical amplitudes, we find that the system can display dynamical multistability if the optomechanical cooperativity exceeds a critical value. This analysis relates standard optomechanical damping to the dynamical attractors known from the theory of optomechanical self-sustained oscillations. We also investigate the effect of thermal and quantum noise and estimate the noise-induced switching rate between the stable states of the system. We then consider applications of this system and primarily focus on how it can be used as bifurcation amplifiers for the detection of small mechanical or optical signals. Finally, we show that in a related but more complicated setup featuring resonant optomechanical interactions, the same effects can be realized with a relaxed requirement on the size of the mechanical oscillations.

• Børkje, Kjetil (2016). Heterodyne photodetection measurements on cavity optomechanical systems: Interpretation of sideband asymmetry and limits to a classical explanation. Physical Review A. Atomic, Molecular, and Optical Physics (PRA). ISSN 1050-2947. 94(4). doi: 10.1103/PhysRevA.94.043816. Full text in Research Archive
• Underwood, M.; Mason, D; Lee, D; Xu, H; Jiang, L. & Shkarin, A.B. [Show all 9 contributors for this article] (2015). Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime. Physical Review A. Atomic, Molecular, and Optical Physics (PRA). ISSN 1050-2947. 92(061801). doi: 10.1103/PhysRevA.92.061801. Full text in Research Archive
• Børkje, Kjetil (2014). Scheme for steady-state preparation of a harmonic oscillator in the first excited state. Physical Review A. Atomic, Molecular, and Optical Physics (PRA). ISSN 1050-2947. 90(023806). doi: 10.1103/PhysRevA.90.023806. Full text in Research Archive
• Ojanen, Teemu & Børkje, Kjetil (2014). Ground-state cooling of mechanical motion in the unresolved sideband regime by use of optomechanically induced transparency. Physical Review A. Atomic, Molecular, and Optical Physics (PRA). ISSN 1050-2947. 90(013824). doi: 10.1103/PhysRevA.90.013824. Full text in Research Archive
• Børkje, Kjetil; Nunnenkamp, Andreas; Teufel, John D. & Girvin, S.M. (2013). Signatures of Nonlinear Cavity Optomechanics in the Weak Coupling Regime. Physical Review Letters. ISSN 0031-9007. 111(053603). doi: 10.1103/PhysRevLett.111.053603. Full text in Research Archive
• Nunnenkamp, Andreas; Børkje, Kjetil & Girvin, S.M. (2012). Cooling in the single-photon strong-coupling regime of cavity optomechanics. Physical Review A. Atomic, Molecular, and Optical Physics (PRA). ISSN 1050-2947. 85(051803(R)). doi: 10.1103/PhysRevA.85.051803. Full text in Research Archive
• Børkje, Kjetil & Girvin, S.M. (2012). Quantum optomechanics with a high-frequency dilational mode in thin dielectric membranes. New Journal of Physics. 14(085016). doi: 10.1088/1367-2630/14/8/085016. Full text in Research Archive
• Jayich, Andrew; Sankey, Jack; Børkje, Kjetil; Lee, Donghun; Yang, Cheng & Underwood, M. [Show all 10 contributors for this article] (2012). Cryogenic optomechanics with a Si3N4 membrane and classical laser noise. New Journal of Physics. 14(115018). doi: 10.1088/1367-2630/14/11/115018. Full text in Research Archive
• Walter, Stefan; Schmidt, Thomas L.; Børkje, Kjetil & Trauzettel, Björn (2011). Detecting Majorana bound states by nanomechanics. Physical Review B. Condensed Matter and Materials Physics. ISSN 1098-0121. 84(224510). doi: 10.1103/PhysRevB.84.224510. Full text in Research Archive
• Børkje, Kjetil; Nunnenkamp, Andreas & Girvin, S.M. (2011). Proposal for Entangling Remote Micromechanical Oscillators via Optical Measurements. Physical Review Letters. ISSN 0031-9007. 107(123601). doi: 10.1103/PhysRevLett.107.123601. Full text in Research Archive
• Nunnenkamp, Andreas; Børkje, Kjetil & Girvin, S.M. (2011). Single-photon Optomechanics. Physical Review Letters. ISSN 0031-9007. 107(063602). doi: 10.1103/PhysRevLett.107.063602. Full text in Research Archive
• Nunnenkamp, Andreas; Børkje, Kjetil; Harris, J.G.E. & Girvin, S.M. (2010). Cooling and squeezing via quadratic optomechanical coupling. Physical Review A. Atomic, Molecular, and Optical Physics (PRA). ISSN 1050-2947. 82(021806(R)). doi: 10.1103/PhysRevA.82.021806. Full text in Research Archive
• Børkje, Kjetil; Nunnenkamp, Andreas; Zwickl, B. M.; Yang, Cheng; Harris, J.G.E. & Girvin, S.M. (2010). Observability of radiation-pressure shot noise in optomechanical systems. Physical Review A. Atomic, Molecular, and Optical Physics (PRA). ISSN 1050-2947. 82(013818). doi: 10.1103/PhysRevA.82.013818. Full text in Research Archive
• Børkje, Kjetil & Sudbø, Asle (2008). Effective theory of fluctuating orbital currents in high-Tc cuprate superconductors. Physical Review B. Condensed Matter and Materials Physics. ISSN 1098-0121. 77(092404). doi: 10.1103/PhysRevB.77.092404. Full text in Research Archive Show summary

  We derive an effective dissipative quantum field theory for fluctuating orbital currents in clean CuO2 sheets of high-Tc cuprates based on a three-band model. The Coulomb repulsion term between Cu and O sites is decoupled in terms of current operators representing horizontal and vertical parts of circulating currents within each CuO2 unit cell of the lattice. The model has ordering of currents at finite temperatures. The dissipative kernel in the model is of the form ||/|q|, indicating Landau damping. Applications of the effective theory to other models are also discussed.

• Børkje, Kjetil (2007). Using Josephson junctions to determine the pairing state of superconductors without crystal inversion symmetry. Physical Review B. Condensed Matter and Materials Physics. ISSN 1098-0121. 76(18). doi: 10.1103/PhysRevB.76.184513. Full text in Research Archive
• Børkje, Kjetil & Sudbø, Asle (2006). Tunneling between noncentrosymmetric superconductors with significant spin-orbit splitting studied theoretically within a two-band treatment. Physical Review B. Condensed Matter and Materials Physics. ISSN 1098-0121. 74. doi: 10.1103/PhysRevB.74.054506. Full text in Research Archive Show summary

  Tunneling between noncentrosymmetric superconductors with significant spin-orbit splitting is studied theoretically in a two-band treatment of the problem. We find that the critical Josephson current may be modulated by changing the relative angle between the vectors describing absence of inversion symmetry on each side of the junction. The presence of two gaps also results in multiple steps in the quasiparticle current-voltage characteristics. We argue that both these effects may help to determine the pairing states in materials like CePt3Si, UIr, and Cd2Re2O7. We propose experimental tests of these ideas, including scanning tunneling microscopy.

• Børkje, Kjetil; Kragset, Steinar & Sudbø, Asle (2005). Instanton correlators and phase transitions in two- and three-dimensional logarithmic plasmas. Physical Review B. Condensed Matter and Materials Physics. ISSN 1098-0121. 71(085112). doi: 10.1103/PhysRevB.71.085112. Full text in Research Archive Show summary

  The existence of a discontinuity in the inverse dielectric constant of the two-dimensional (2D) Coulomb gas is demonstrated on purely numerical grounds. This is done by expanding the free energy in an applied twist and performing a finite-size scaling analysis of the coefficients of higher-order terms. The phase transition, driven by unbinding of dipoles, corresponds to the Kosterlitz-Thouless transition in the 2D XY model. The method developed is also used for investigating the possibility of a Kosterlitz-Thouless phase transition in a three-dimensional system of point charges interacting with a logarithmic pair-potential, a system related to effective theories of low-dimensional strongly correlated systems. We also contrast the finite-size scaling of the fluctuations of the dipole moments of the two-dimensional Coulomb gas and the three-dimensional logarithmic system to those of the three-dimensional Coulomb gas. ©2005 The American Physical Society

• Brevik, Iver Håkon; Børkje, Kjetil & Morten, Jan Petter (2004). Two-brane Randall-Sundrum model in AdS(5) and dS(5). General Relativity and Gravitation. ISSN 0001-7701. 36, p. 2021–2038. Full text in Research Archive

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• Børkje, Kjetil (2025). Kan et lite kjøleskap bli nøkkelen til kvante­data­maskinens gjennom­brudd? [Journal]. forskning.no. Full text in Research Archive
• Børkje, Kjetil (2025). Multipartite entanglement witnesses from photodetection statistics. Full text in Research Archive
• Børkje, Kjetil (2024). Nonclassicality in optomechanics from single sideband-photon detection: beyond the single phonon regime. Full text in Research Archive
• Børkje, Kjetil (2023). Nonclassical photon statistics in two-tone continuously driven optomechanics. Full text in Research Archive
• Børkje, Kjetil (2022). Signatures of nonclassicality in optomechanical systems. Full text in Research Archive
• Børkje, Kjetil (2020). Can quantum physics help improve sensing and imaging? Full text in Research Archive
• Børkje, Kjetil (2020). Critical quantum fluctuations in optomechanical systems with large single-photon cooperativity. Full text in Research Archive
• Børkje, Kjetil & Johansen, Lars M. (2020). En milepæl i utviklingen av kvantedatamaskiner. 13(02), p. 92–93. Full text in Research Archive
• Børkje, Kjetil (2019). Quantum sensing and mechanical systems 
in the quantum regime. Full text in Research Archive
• Børkje, Kjetil (2019). Quantum nonlinear dynamics in a 
multimode optomechanical system with 
large single-photon cooperativity. Full text in Research Archive
• Børkje, Kjetil (2019). Critical quantum fluctuations in optomechanical systems with large single-photon cooperativity. Full text in Research Archive
• Børkje, Kjetil (2019). Utilizing the nonlinear dynamical response of an optically damped mechanical oscillator. Full text in Research Archive
• Børkje, Kjetil (2018). Dynamical bistability and bifurcation-based amplification with two-mode optomechanics. Full text in Research Archive
• Børkje, Kjetil (2017). Kvanteteknologi: Når science fiction blir til virkelighet. Full text in Research Archive
• Børkje, Kjetil (2017). Interpretation of sideband asymmetry in optomechanics. Full text in Research Archive
• Børkje, Kjetil (2017). Quantum optomechanics. Full text in Research Archive
• Børkje, Kjetil (2017). Single-photon optomechanics. Full text in Research Archive
• Børkje, Kjetil (2015). Testing quantum mechanics at large scales – for applications and for fun. Full text in Research Archive
• Børkje, Kjetil (2014). Ground state laser cooling of low-frequency mechanical motion. Full text in Research Archive
• Børkje, Kjetil (2014). Motivations for Increasing the Single-Photon Coupling Strength in Cavity Optomechanical Systems. Full text in Research Archive
• Børkje, Kjetil (2006). Physics of non-centrosymmetric superconductors with significant spin-orbit splitting. Full text in Research Archive
• Børkje, Kjetil (2006). Tunneling between noncentrosymmetric superconductors. Full text in Research Archive
• Børkje, Kjetil (2008). Theoretical Studies of Unconventional Order in Quantum Many-Particle Systems. Norges teknisk-naturvitenskapelige universitet. ISSN 9788247163252. Full text in Research Archive

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