Coherence of quantum non-Gaussian states via nonlinear absorption of quanta

Quantum Science and Technology PAPER • OPEN ACCESS Coherence of quantum non-Gaussian states via nonlinear absorption of quanta To cite this article: Kingshuk Adhikary et al 2025 Quantum Sci. Technol. 10 035048 View the article online for updates and enhancements. You may also like Corrigendum: Urban precipitation dynamics risk assessment across Indian smart cities (2025 Environ. Res. Lett. 20 074059) Vijay Jain, Shekhar Singh and Manish Kumar Goyal - Erratum: Influence of Finite Diffusion on Cation Insertion-Coupled Electron Transfer Kinetics in Thin Film Electrodes [J. Electrochem. Soc., 171, 010527 (2024)] Matthew Chagnot, Sofia Abello, Ruocun Wang et al. - Erratum: Improvement of light extraction and carrier injection in far UV-C LEDs on AlN substrate via conductive pAl0.45Ga0.55N contact layer Appl. Phys. Express 18, 031003 (2025) Hirotsugu Kobayashi, TaeGi Lee, Yusuke Okuaki et al. - This content was downloaded from IP address 158.194.85.200 on 24/11/2025 at 12:53 Quantum Sci. Technol. 10 (2025) 035048 https://doi.org/10.1088/2058-9565/ade334 OPEN ACCESS RECEIVED 3 October 2024 REVISED 31 January 2025 ACCEPTED FOR PUBLICATION 10 June 2025 PUBLISHED 26 June 2025 Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. PAPER Coherence of quantum non-Gaussian states via nonlinear absorption of quanta Kingshuk Adhikary, Darren W Moore∗and Radim Filip Department of Optics, Palack´ y University, 17. listopadu 1192/12, 779 00 Olomouc, Czech Republic ∗Author to whom any correspondence should be addressed. E-mail: [email protected] Keywords: quantum coherence, quantum non-Gaussian, quantum engineering and technology Abstract The linear and phase insensitive absorption of a single quanta via coherent interactions with a saturable system, even a single ground state qubit, is sufficient to deterministically generate quantum non-Gaussian states in an oscillator, even stimulated merely by increasing thermal oscillator energy. However, the resultant states only approach Fock states and therefore do not exhibit quantum coherence. Here we overcome this limitation using a minimal step: a nonlinear phase-insensitive absorption process added to the linear one. The coherent addition of such individually passive processes allows coherence to emerge and increase in phase space without an external drive and with minimal interaction requirements. The coherence of quantum non-Gaussian states emerges because the linear and nonlinear absorption processes are not mutually passive. In the simplest case rotationally symmetric Wigner functions of the oscillator Fock states convert their many negative regions to an extremely complex asymmetric structure in sharp contrast to the rotational symmetry of those obtained by the individual interactions. We extend this case to include an unsaturable absorber (oscillator) and analyse switching between linear and nonlinear absorptions, suitable for broad classes of experiments. 1. Introduction A core property of quantum theory is the capacity of an individual system to form coherent superpositions [1,2]. While every quantum state is a superposition in some basis, some superpositions in particular contexts are exceptionally fundamental or relevant for applications. Quantum coherence undergoes intensive research in a large variety of contexts, such as open systems where only certain superpositions survive decoherence [3,4] or where observable coherences are generated via external driving [5,6], or as a resource in quantum technologies [7,8] and quantum thermodynamics [9]. Especially fundamental and relevant are superpositions of energy eigenstates and the ways in which they can arise. They already have diverse and experimentally demonstrated applications in quantum phase sensing [7,10] and bosonic quantum error correction [11,12] for quantum computing and communication. Surprisingly, basic quantum interactions deterministically generate quantum non-Gaussian oscillator states via linear coherent absorption of quanta from a thermal oscillator [13,14]. Specifically, phase insensitive and energy conserving absorption of energy from a thermal oscillator by a ground state qubit unconditionally generates states that can approach Fock states, without any external processes such as measurement, feedforward or driven/dissipative engineering. However, they cannot generate local oscillator coherence even if they do generate light-matter entanglement [15]. Indeed this limitation is a general property of multiphoton Jaynes–Cummings (JC) models [16] (see figure 1). In this paper we overcome this limitation using a nonlinear phase-insensitive absorption added to the linear one [13,14] in a fully quantum mechanical way. Thus, a combination of two different phase-insensitive absorption processes, for simplicity tested on Fock states, individually energy conserving © 2025 The Author(s). Published by IOP Publishing Ltd Quantum Sci. Technol. 10 (2025) 035048 K Adhikary et al Figure 1. Coherence of quantum non-Gaussian states via a combination of linear and nonlinear absorption by a single qubit: A ground state qubit absorbing fixed quanta of energy (one or two in this illustration) from an oscillator prepared in a Fock state produces only mixtures of Fock states. These states are rotationally symmetric in phase space and therefore the coherence strictly vanishes. In contrast, if both interactions are simultaneous then a superposition of absorptions results which begins to break the rotational symmetry, indicating the emergence of quantum coherence. Wigner function transformations: The transformation of the Wigner function of Fock state |2⟩remains an incoherent mixture of Fock states for the individual interactions. The combination however results in the emergence of coherence, C=0.08 and the loss of rotational symmetry. The Wigner function in the figure is found for interaction strengths g(2) g(1)=0.1 and short interaction time τ=0.157. and incapable of producing coherence, jointly results in coherent quantum non-Gaussian states. The combination is essential to create frustration between the conditions required for the interaction to be passive, i.e. energy conserving, thus allowing oscillator superpositions to deterministically develop even after ignoring the final state of the qubit. The resultant superpositions in the Fock basis show substantial quantum coherence and quantum non-Gaussian features, retaining the Wigner negativity of the original Fock states [13,14] (see figure 1). In what follows we demonstrate the striking extent to which this apparently simple compund interaction generates extremely quantum non-Gaussian states with substantial coherence, compare it to the classical incoherent oscillator states, extend this idea to an oscillator absorber, and suggest a feasible experiment to verify the emergence of coherence from nonlinear absorption. We close with some discussion of the nature of the nonlinear absorption interaction with respect to coherence generation. 2. Results To advance the results in [13,14], we address the emergence of local oscillator coherence via absorption of energy from the oscillator, starting from the pure incoherent Fock states approached by those methods. The relevant basis in which to examine nontrivial superpositions is therefore the energy eigenbasis of the oscillator, given by the Fock states |n⟩, the eigenstates of the harmonic oscillator Hamiltonian Hω=ωb†b, with constant frequency ω. Naturally, the free evolution of the oscillator does not create superpositions from such a setup so the oscillator must interact with a new subsystem with free evolution HΩ, itself prepared in an incoherent state, diagonal in the energy eigenbasis. We will typically take this to be the ground state of HΩ, which can be approached by cooling. For a qubit subsystem we have HΩ=Ω 2σz, with σza Pauli matrix. Let us now be more precise. To quantify the overall coherence for an oscillator we use the relative entropy of coherence [17] defined as C(ρ) = S(ρdiag)−S(ρ),(1) where Sis the von Neumann entropy and ρdiag is the diagonal matrix containing the principal diagonal of the corresponding density matrix ρ. For JC-like interactions the oscillator and absorber Hamiltonians which set the energy eigenbases are Hω(above). The kth order JC interaction takes the form V(k)=g(k)(σ+bk+σ−(b†)k),(2) where σ±are the qubit raising and lowering operators. As said, such interactions are known to preserve the local incoherence of incoherent initial states [16]. Indeed the free evolution H0=Hω+HΩcommutes with V(k)for all k, provided certain frequency conditions are met. More precisely, [H0,V(k)]=g(k)(kω−Ω)(σ−(b†)k−σ+bk),(3) which is only zero for Ω = kω. That is, the energy N=kσ+σ−+a†ais a conserved quantity. When the interaction Hamiltonian commutes with the free Hamiltonian no local oscillator coherence emerges. We 2 Quantum Sci. Technol. 10 (2025) 035048 K Adhikary et al note in passing that a detuned model, with Hamiltonian H=−∆a†a+Ω 2σz+V(k)and frequency ∆, still does not generate local oscillator coherence. A simple method to overcome this limitation and produce oscillator coherences is to combine, in a fully quantum way, two of these energy conserving interactions (equation (2)) with different k. 2.1. Short-time emergence of coherence and sequential approach Let’s examine the simplest case of k=1 and k=2, so that the interaction has the form V=V(1)+V(2). This combines linear (k=1) and nonlinear (k=2) absorption by the qubit. Apart from simplicity this is also the most relevant experimentally, as it involves the already extremely well characterised lowest order JC interactions [18]. For such interactions the number of excitations is no longer conserved, that is, there is frequency frustration between the competing absorption processes. Despite the analytically intractable nature of the system, some insight into the emergence of coherent quantum non-Gaussian states can be gained by considering the short time evolution of the system. An illuminating approach is to consider the first order expansion of the unitary evolution via the Baker-Campbell-Hausdorff theorem. That is, to first order in t≪1 we have U=e−i(V(1)+V(2))t=e−iV(1)te−iV(2)t+O(t2). Terms beyond this approximation further increase the coherence. For this approximation, coherent quantum non-Gaussian states already emerge. Advantageously, this approximation also motivates Hamiltonian switching between interactions V(1) and V(2)or vice versa. This sequential method provides an alternative and immediately accessible procedure to produce coherence by combining passive and phase insensitive interactions coherently absorbing individual quanta. For example, in the context of trapped ions such interactions are generated by illuminating the ion at the k−th sideband. Therefore implementation of the switching protocol requires only that two such sidebands are independently available and controllable in the same setup [7]. In figure 2we show the emergence of coherence at the level of the Wigner functions. Starting with the V(1)interaction and the state |g⟩|n⟩≡|g,n⟩,n>0, the states |g,n⟩and |e,n−1⟩become coupled, and the typical state is a superposition of these two. Tracing out the qubit does not produce any oscillator coherence. When the Hamiltonian is switched to V(2), these two states decouple and couple to new states: |g,n⟩couples to |e,n−2⟩, and |e,n−1⟩couples to |g,n+1⟩. A typical state is now a superposition of these four, and the average total number of excitations has changed. In fact, we can explicitly write the state as |Ψ1⟩=|g⟩(α(t)|n⟩+β(t)|n+1⟩) + |e⟩(γ(t)|n−1⟩+δ(t)|n−2⟩)(4) where α(t) = cos(g(2)√n(n−1)t)cos(g(1)√nt)(5) β(t) = −sin(g(2)√n(n+1)t)sin(g(1)√nt)(6) γ(t) = −icos(g(2)√n(n+1)t)sin(g(1)√nt)(7) δ(t) = −isin(g(2)√n(n−1)t)cos(g(1)√nt).(8) When the qubit is now traced out the remaining oscillator is typically in a superposition of Fock states. Indeed each qubit eigenstate is coupled to a nonoverlapping superposition of Fock states so that the energy of the qubit no longer specifies the energy of the oscillator. Similar analyses hold for the inverted order of sequential operations, albeit with a different constraint on initial n. The Wigner function corresponding to the maximum coherence for the sequential coherence emergence is shown in figure 2. The result is a maximum qubit coherence of C ≈ln2 while the Wigner function loses rotational symmetry. Equation (4) shows that tracing out the qubit results in a mixture of superpositions, each from a two dimensional subspace. Due to this structure the coherence is not increased by increasing n, in contrast with what follows in the long time emergence of coherence. Thus, the coherence of the oscillator in equation (4) is indeed bounded by the two dimensional subspace. However, coherence only mildly increases when repeating the switching procedure. Furthermore varying the individual interaction times for each step does not increase the coherence, and changing the order to start with V(2)decreases the ranges of times for which the maximum coherence emerges. We now move to the more autonomous dynamics without switching, where these limitations are surpassed. 3 Quantum Sci. Technol. 10 (2025) 035048 K Adhikary et al Figure 2. Sequential emergence of coherent quantum non-Gaussian states, C=0.7 by two sequential linear and nonlinear absorptive operations. The first step, linear absorption, prepares entanglement between qubit and the oscillator but fails to produce coherence in the oscillator (see Wigner function). After the second step with a nonlinear absorption coherence already emerges and the rotational symmetry is strongly broken for higher Fock occupations. The initial state is the Fock state |7⟩, the interaction time is t=1.57, the same for both steps, and the remaining detailed parameters are given in figure 3. Figure 3. The emergence of coherent quantum non-Gaussian states from frequency frustrated nonlinear absorption, stimulated by initially incoherent Fock states in bat a fixed coupling ratio g(2) g(1)=0.1. The initial (τ=0) highly non-Gaussian state is the Fock state |7⟩. The rightmost state corresponds to the maximum coherence, C≈4, achieved over the interval 0 ⩽τ⩽2πat time τ=3.32 and more than 4 times larger than the short time approximation discussed in the main text. The central states correspond to an example of a state with half the maximum coherence, in this case achieved at τ=0.95. The states remain radically non-Gaussian, containing many negative regions and rotational symmetry is completely lost. The corresponding density matrices, with entries ρnm, below the Wigner functions show that Vtends to generate states with superpositions between large and small Fock states with entries very far from the original Fock state, and only small contributions from the ground state. More details on the parameter choices are given in the appendix. 2.2. Long-time emergence of coherence Figure 3shows typical examples of the oscillator at several stages after the compound interaction simultaneously involving V(1)and V(2). The initial state is always taken to be the incoherent state |g⟩|n⟩, where |g⟩is the ground state of the absorber and |n⟩is a Fock state of the oscillator. The ratio of coupling strengths is set to g(2) g(1)=0.1, where the linear absorption still dominates, over the range 0 ⩽τ⩽2πwhere τ=g(2)tis a scaled time. Details on these parameter choices for the coherence dynamics are given in an appendix. Once frustration of the energy conservation conditions is introduced via the combined linear and nonlinear absorption processes substantial coherence is gradually generated alongside strikingly complex Wigner distributions with strongly broken rotational symmetry and large density matrix coherences (off-diagonal elements of the density matrices). As the Wigner functions develop, their prominent negative regions persist despite the mixedness introduced by the tracing out of the resonant absorber. That is, the states produced do not belong to the class of states defined by the convex mixture of Gaussian states. States from this class may be non-Gaussian [19], but they do not possess any Wigner negativity. Importantly, the rotational symmetry is gradually broken in time, as visible in figure 3, and the Wigner function approaches a completely new topology in phase space, going even beyond the complexity of those currently measured in nonlinear potentials [20]. The breaking of the rotational symmetry combines classical coherent displacement in phase space with quantum non-Gaussian symmetry breaking of the negative parts of interference effects 4 Quantum Sci. Technol. 10 (2025) 035048 K Adhikary et al Figure 4. The spread into the Fock basis and frustration of energy conservation is captured by the rise in mean energy of the system ⟨N⟩, accompanied by a large increase in the standard deviation ∆N. The maximum coherence occurs at the dashed vertical line. The bar chart shows the maximum coherence generated with g(2) g(1)=0.1 over the range 0 ⩽τ⩽2πas a function of the initial Fock state |n⟩. The maximum coherence generally increases with nup to saturation at n=7. The blue bars indicate the removal of the Gaussian shell via displacement and squeezing operations (detailed in text). The coherence persists and is thus well beyond the covariance matrix approximation. in the Wigner function. Such complex symmetry broken structures appear close to the maximum of the mean number of quanta along with a large increase in the noise. The value of the coherence gives only an overall view of the Fock state superpositions contributing to the coherence. Examining the density matrix coherences in figure 3, they spread deeply into the Fock basis, coupling low and high Fock states. This feature is not captured by the short-time approximation, nor when extended to a sequential scheme where the short-time approximation operators are repeatedly applied. Additionally, there is only a marginal contribution from the ground state. The mean number of quanta produced in the full dynamics (see figure 4) is substantially higher than that of sequential method and the spread into the Fock basis far beyond the initial occupation number is reflected in the growth of the mean energy of the system ⟨N⟩≫7. This effect is already known for linear absorption [21] but here is accompanied by the emergence of coherence. That is, linear absorption can result in an increase in mean energy, if the linear absorption is associated with blue-detuned interactions. However, it also results in a reduction of the noise in energy, as the output states closely approximate Fock states. The addition of nonlinear absorption results in a simultaneous increase in both mean energy ⟨N⟩and noise ∆N, which allows for the emergence of coherence. Figure 4also shows the increase in maximum coherence achieved over the range 0 ⩽τ⩽2πas a function of initial Fock state. There is a notable increase in the maximum achievable coherence with increasing n, up to saturation at n=7. The blue bars show the coherence after the removal of the Gaussian approximation, i.e. displacement and squeezing are applied until the mean values of X=1 √2(b+b†)and P=i √2(b†−b)are zero and the covariance matrix is diagonal with equal entries. Quantum coherence due to Gaussian displacement/squeezing is thus removed indicating that the coherence beyond the Gaussian approximation is substantial. A negative Wigner function remains negative under Gaussian operations thus the coherence is strongly connected to the quantum non-Gaussianity of the state. 3. Discussion 3.1. Weak coupling regimes, dephasing and classical initial states To extend this result to many possible experimental scenarios, we analyse two significant potential obstacles in even well-isolated oscillators: the presence of free evolution alongside the compound interaction and external dephasing processes. Thus far, for simplicity, these discussions have taken place in the ultra-strong coupling regime, in which the free motion can be neglected. Reintroducing the free motion adds significant complexity to an already intractable problem. However we are interested in the emergence of coherence, rather than its optimisation. Therefore we compare the maximum coherence when the free motion is relevant with the maximum coherence obtained from our example in figure 3, keeping the remaining parameters unchanged. Below the saturation observed at n=7 it is possible to find regions outside the 5 Quantum Sci. Technol. 10 (2025) 035048 K Adhikary et al ultrastrong coupling regime where the coherence can be enhanced. This reflects a similar finding for qubit coherence in the Rabi model emerging from a similar unitary setting [22]. That is, this remarkable emergence of coherence is not restricted to the experimentally challenging ultrastrong coupling regime, but is much more common and may even be greater outside it. Above n=7 it is typical for the maximum coherence to be in the ultrastrong coupling regime. However even when the coherence is lowered the free motion does not significantly impact the quantum non-Gaussianity or complexity and negativity of the resulting Wigner functions (examples in appendix). Similarly, the density matrices still show a substantial spread into the Fock basis coherences. As expected, coupling to a dephasing environment strongly reduces the coherence. We give an example in the appendix where quantum non-Gaussian features confirmed by negativity of the Wigner function appear at shorter times but are eventually suppressed by the decoherence process, despite leaving a non-Gaussian state with nontrivial coherence. One may wonder if the emergence of coherence from nonlinear absorption is due to the nonclassical features of the Fock states or of the saturability of the qubit absorber, both of which we have relied on throughout. In fact the qualitative features of our results hold for initial states which are classical mixed states showing only thermal noise in the excitation number, as well as when the qubit is replaced by an unsaturable oscillator (see appendix). Strikingly, the negative features of the Wigner function are more robust to initial thermal noise than to decoherence. 3.2. Extension of frustration of energy conservation to other cases To create coherence in a single oscillator from an incoherent state the oscillator energy must change and the interaction Hamiltonian must not commute with the oscillator free motion, [HΩ,V]=0. With a total Hamiltonian H=Hω+HΩ+V, there are two distinct possibilities. Either (i) [Hω+HΩ,V] = 0 or (ii) [Hω+HΩ,V]=0. For the first case, it follows that [HΩ,V] = −[Hω,V]=0. In this case sum of the energies of the subsystems is conserved, so that overall Vdescribes a globally passive process. In this case, even though the local excitation number of the oscillator can change, no coherence emerges. Since the total excitation number is conserved, any change in the energy of subsystem Hωis directly compensated for by gain or loss of energy in subsystem HΩ. That is, energy exchange between the subsystems can be mediated passively by the interaction V, without any net exchange of energy stored in the interaction, so that there is no uncertainty in the oscillator energy. This explains why phase insensitive interactions such as JC, beamsplitters, or even trilinear interactions do not produce oscillator coherence. Passive interactions do not produce coherence in the oscillator and this holds even when the interaction is locally active. For the second case there are two subcases: (a) [HΩ,V] = 0 and (b) 0 = [HΩ,V]=−[Hω,V]. For case (a) it is still possible to generate coherence. For example, the optomechanical interaction a†a(b+b†)[23] will generate coherence in the mechanical mode beven though the optical mode’s free Hamiltonian commutes with the interaction and similarly for the dispersive Rabi interaction σz(b+b†)in superconducting circuits [24–27] and spin-mechanics [28]. This occurs even if the optical or qubit systems are not prepared in the ground state. Since their energy remains constant, yet the oscillator mode gains or loses energy, there must be an active contribution from the interaction. For the two cases above, it comes because of the counterrotating terms in the interaction; however, for our case here, we combine interactions which are each separately in the rotating wave approximation. This becomes even more appealing in case (b), which contains the nonlinear absorption interaction studied in this manuscript; moreover as with the optomechanical and dispersive Rabi interactions the nonlinear absorption method does not need to depend on the saturability of the absorber (see appendix), as in our simple example of equation (2). Additionally our interaction is not limited to the simplest case we selected involving k=1 and k=2. Any pair of kwill continue to be contained in case (b), and produce coherent quantum non-Gaussian states. The frustration of the conditions for commutation to hold prevents both subsystems from conserving their energy in a nontrivial way: allowing only mutually active transformations that may result in oscillator coherence. 4. Conclusion Clearly, non-Gaussian quantum superpositions in oscillators require certain minimal conditions to be met in order to arise without initial coherence, direct external coherent driving of the subsystems and counterrotating terms in the interaction Hamiltonians. Here we have used experimentally feasible phase-insensitive interactions to demonstrate some of these required conditions. That is, for this minimal case, with all subsystems prepared in incoherent states, the resulting evolution must involve a mutually active transformation in order for quantum coherence to emerge. This is not a sufficient condition, but a necessary one, so any particular Hamiltonian used like this must also be thoroughly explored for such effects. 6 Quantum Sci. Technol. 10 (2025) 035048 K Adhikary et al The necessity of a mutually active transformation implies, through conservation of energy, that another physical system is at least effectively present and donating or receiving energy from the oscillator (see appendix for discussion). In many recent cases of quantum technology, this extra physical system is in fact the external drive present during state generation that we have avoided throughout the discussion. Crucially, there are many systems in which an effective Hamiltonian dynamics can be derived which allows this source/sink of energy to be fully externalised. The effective Hamiltonian obscures the origin of quantum coherence which may require the completion of the full Hamiltonian as outlined in the appendix. Instead of searching for systems which can be externalised in this way, one may look to naturally occurring forces or technological arrangements of matter whose internal structure contains the required energy source/sink to generate coherence without external drives or counterrotating terms during the state generation. Such forces or technology may then provide a minimal approach to reservoir engineering, in which the internal structure replaces the externally driven engineered environment. This alternative transient method is a new starting point in reservoir engineering methods. Again, and in contrast to them, this method does not contain either counter-rotating terms or external coherent drives during the state preparation applied to incoherent states [29]. Continuing from the starting point, such minimal mechanisms can be extended for the generation of quantum non-Gaussian states without the above mentioned tools typically used in superconducting circuits [25,30], trapped ions [31], optomechanical systems [32,33], and two and multi-mode non-Gaussian entanglement clearly distinguishable from previously analysed cases [34]. Searching for such possibilities beyond spin-mechanical interactions with counterrotating terms [28,35], and using modern technology with quantum systems may open many exciting doors in various quantum technologies requiring coherent quantum non-Gaussian states [36–40]. The high-quality Fock states used to initiate these effects are routinely available for trapped ions and superconducting circuits [21,41]. However, they can also be obtained with a purely linear coherent absorption process within the rotating wave approximation [13,14], and are thus available for testing these minimal conditions. These processes tend to result in the required Fock state in an admixture with the ground state. Even for very high contamination this does not prevent the emergence of coherence or the quantum non-Gaussian features we have discussed (see appendix). Moreover we are not limited to such Fock states, or imperfect versions thereof, as a direct observation of these effects can also emerges from initially incoherent thermal or Poissonian oscillator statistics (see appendix) which are also readily prepared in systems such as trapped ions and superconducting circuits. Data availability statement The data that support the findings of this study are openly available at the following URL/DOI: https:// zenodo.org/records/15392092 [42]. Acknowledgment The authors acknowledge funding from Project No. GA22-27431S of the Czech Science Foundation and the project CZ.02.01.01/00/22_008/0004649 (QUEENTEC) of EU and MEYS Czech Republic. R.F. was also supported by the European Union’s HORIZON Research and Innovation Actions under Grant Agreement no. 101080173 (CLUSTEC) and the Quantera project CLUSSTAR (8C24003) of MEYS Czech Republic. Project CLUSSTAR has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under Grant Agreement No. 731473 and 101017733 (QuantERA). Appendix A. Sequential method We expand on the sequential scheme proposed in section 2.1. The short time approximation using the first terms of the BCH theorem is a first step which already produces coherence and breaks total excitation number conservation. First, we note the quantum non-Gaussian states that emerge from this dynamics and compare with the main result in figure 3. In figure 5we show the effect of this Hamiltonian switching with V(2)initiating, n=7, and g(2) g(1)=0.1. Repeated switching does not substantially increase the available coherence, nor does increasing the initial Fock state occupation. 7 Quantum Sci. Technol. 10 (2025) 035048 K Adhikary et al Figure 5. An example of the Hamiltonian switching process using V(2)to initiate the qubit-oscillator entanglement. Each interaction has the same time interval t, the initial oscillator occupation is n=7 and the ratio of coupling strengths is again g(2) g(1)=0.1. Figure 6. Coherence as a function of time for several coupling strength ratios and initial Fock states. From left to right, g(2) g(1)=10,1,0.1. For relatively large g(2)the coherence dynamics has an oscillatory character, which is lost for lower g(2)but with a large gain in achievable coherence. The saturation with increasing nis already visible here. Appendix B. Coherence dynamics The coherence for the nonlinear absorption has a complex time evolution and dependence on the initial state and coupling strengths, displayed in figure 6. In the main text we have selected g(2) g(1)=0.1 as it produces very large coherence. The oscillatory behaviour is lost, but the magnitude of the coherence is greatly enhanced by this choice. The maximum coherence generally appears at τ=π. When this interaction time is fixed, it becomes clear that the maximum coherence tends to occur around g(2) g(1)=0.1 as nincreases, as in figure 7. Although some higher values of ndeviate from this, the increase in coherence compared to this choice of coupling strengths is negligible, as can be seen by comparing figures 7and 6. After fixing this choice, the maximum values of coherence used in the main text are those optimised over the time interval 0 ⩽τ⩽2π. Figure 7also shows the times at which these maximum coherences occur as a function of the initial Fock occupation n. Although there are fluctuations due to minor changes in maximum coherence, this confirms the intuition that the maximum occurs around τ=π. Appendix C. Weak coupling and decoherence Here we give greater details on the points made in the first subsection from the discussion. Firstly, outside the ultrastrong coupling regime the effects of the free motion terms of the Hamiltonian are relevant to the coherence dynamics. As discussed in the main text the effects we have described are not limited to regimes with such large interaction strengths. Figure 8shows the Wigner function and density matrix of the maximum coherence for n=7 and g(2) g(1)=0.1 with ω=Ω=1, resulting in C=3.5. This well approximates the maximum coherence achieved in the ultrastrong coupling regime but produces quite different output states. Nevertheless the qualitative features remain: large coherence, superpositions between distant Fock states, and complex Wigner functions with multiple negative regions. 8