Nonperturbative electromagnetic nonlinearities, n-photon reflectors, and Fock-state lasers based on deep-strong coupling of light and matter

Light and matter can now interact in a regime where their coupling is stronger than their bare energies. This deep-strong coupling (DSC) regime of quantum electrodynamics promises to challenge many conventional assumptions about the physics of light and matter. Here, we show how light and matter interactions in this regime give rise to electromagnetic nonlinearities dramatically different from those of naturally existing materials. Excitations in the DSC regime act as photons with a linear energy spectrum up to a critical excitation number, after which, the system suddenly becomes strongly anharmonic, thus acting as an effective intensity-dependent nonlinearity of an extremely high order. We show that this behavior allows for N-photon blockade (with $N \gg 1$), enabling qualitatively new kinds of quantum light sources. For example, this nonlinearity forms the basis for a new type of gain medium, which when integrated into a laser or maser, produces large Fock states (rather than coherent states). Such Fock states could in principle have photon numbers orders of magnitude larger than any realized previously, and would be protected from dissipation by a new type of equilibrium between nonlinear gain and linear loss. We discuss paths to experimental realization of the effects described here.

Recent successes in the coupling of matter and light now make it possible to realize regimes of light-matter interactions in which the coupling between light and matter can be much stronger than in established optical technologies.Because of the central role the physics of light and matter plays in many fields, these new coupling regimes are being intensely explored.One such example is the ultra-strong coupling regime, where the coupling energy is within an order of magnitude of the bare energies of the light and matter [1].Such regimes promise to give rise to new chemical processes [2][3][4][5], strong modifications of transport and thermodynamic properties of materials [6,7], new phases of matter, quantum simulators, and quantum technologies more broadly [1,8].
Taking these ideas to their logical extreme is the so-called deep-strong coupling regime (DSC), where the strength of the coupling is greater than the bare energies of the light and matter.In the past few years, the first experiments in this regime have emerged [9,10].Much of the interest in ultra-strong and deep-strong coupling is focused on the properties of the ground state of either one or many emitters coupled to a cavity mode, leading to many interesting new phenomena such as light-matter decoupling [11,12], population collapses and revivals [13], large Lamb shifts leading to inversion of qubit energy levels [9,14], and renormalization of qubit energy levels by a photonic continuum [10].Likely, many of the potential applications of this regime have yet to be identified.
Here, we consider the opportunities afforded to us by the excited states of a DSC system, which are important from the perspective of quantum and nonlinear optics.For example, the emission of light in such systems probes the excited states.First, we show that deep-strong coupling of a two-level system to a resonant cavity leads to the formation of excitations ("photonic quasiparticles" [15], which we refer to as "DSC photons") with nonlinear properties much different than those in any known system.Then, we analyze the coupling of an emitter to this nonlinear photonic quasiparticle.We find that the coupling of an excited two-level system to this nonlinear system enables a phenomenon of N -photon blockade in which N excitations can be populated, but N + 1 cannot.We show that a laser or maser based on stimulated emission of DSC photons behaves fundamentally differently from a conventional maser or laser.Specifically, this maser produces close approximations to Fock states in its steady-state, rather than coherent states, as in conventional lasers.They could have a few hundred photons, thus being orders of magnitude larger than any Fock states realized thus far.Moreover, Fock states produced by this mechanism are stable against dissipation as they arise from a new type of equilibrium between nonlinear gain and linear loss.Our results may thus help to address the long-standing problem in quantum science of generating Fock states.Finally, we discuss how the concept developed here can be implemented in superconducting qubit platforms.

I. NONLINEAR PHOTONIC QUASIPARTICLES BASED ON DEEP STRONG LIGHT-MATTER COUPLING
Fundamental to our results is the spectrum of a twolevel system (qubit) interacting with a single-mode cavity (schematically illustrated in Fig. 1a), which we review here [16].The Hamiltonian, referred to as the (generalized) Rabi Here, λ = 0. (c) Successive excitation energies for a single spin sector for different coupling values.For g ≫ 1 the excitation energies are constant, as for a bare photon.At large photon number, they deviate rapidly and nonlinearly from harmonicity, akin to a photon with a strongly intensity-dependent nonlinearity.
Hamiltonian, is given by Here, ω 0 is the transition frequency of the two-level system, σ x,z are the x and z Pauli matrices, ω is the cavity frequency, a ( †) is the cavity annhilation (creation) operator, and g is the Rabi frequency.It will be convenient to non-dimensionalize the coupling as g = g/ω.We have also generalized the standard Rabi Hamiltonian by including a term λσ x which is relevant in contexts of superconducting qubits with applied bias fluxes [9].For simplicity of presentation, we consider the case of λ = 0, which leads to approximately degenerate spin states (and in which case the qubit frequency is ω 0 ).In the SI, and in various numerical results, we do consider the effect of a finite λ term, which yields the same qualitative conclusions.While the Rabi Hamiltonian cannot be analytically diagonalized in general, an approximate spectrum can be found for the regime g ≫ 1, which forms the basis for our analytical theory.In the SI, it is shown that the approximate eigenstates are labeled by an oscillator quantum number n = 0, 1, 2, . . .and a spin quantum number σ = ±1.These eigenstates |nσ⟩ and corresponding energies E nσ , for g ≫ 1, are given by where D(z) ≡ exp z(a † − a) is the displacement operator, and L n is the Laguerre polynomial of order n.The state |n⟩ on the right-hand side refers to the Fock basis of the cavity, while the states |x±⟩ refer to the x-polarized spin states of the qubit.The spectrum is plotted in Fig. 1b (adding an g-dependent offset ℏωg 2 for convenience).As seen in Eq. ( 2), the spectrum in the DSC regime is organized into two oscillator-like ladders (one for each spin).Moreover, for large g, the spectrum appears almost completely harmonic, indicating the existence of an effective photon (or photonic quasiparticle, which we will sometimes call a DSC photon).To understand this, we note that for g ≫ 1, the σ z acts as a perturbation to the remaining Hamiltonian, where b = D † (gσ x )aD(gσ x ) = a + gσ x .This approximate Hamiltonian admits a harmonic spectrum, in which the new oscillator variables b obey canonical commutation relations [b, b † ] = 1, and excitations are constructed by applying further b † operators.In other words, the eigenstates of H DSC are Fock states of b, or equivalently, displaced Fock states of a.
The σ z term breaks the even spacing of the ladder, leading to an anharmonicity (equivalently, nonlinearity) which we now quantify.Without loss of generality, we will focus on the lower-energy σ = −1 ladder, enabling us to omit the spin index in our notation.We assess the "harmonicity" of the spectrum by plotting successive excitation energies E n+1 − E n as a function of n, as in Fig. 1c (in units of ℏω).We will refer to n as the "photon number."For strong and ultrastrong coupling, the spectrum is anharmonic at the level of a single photon, leading to the familiar phenomenon of photon blockade.For deep-strong coupling, the behavior is quite different: the spectrum is harmonic up to some critical excitation number (n c ∼ g 2 ), and then rapidly becomes anharmonic.This may be seen directly from the properties of L n (x).
To understand the relation of this strong anharmonicity to existing nonlinear optical systems, recall that a singlemode cavity, with a Kerr nonlinear medium inside of it, can be described by a Hamiltonian of the form H Kerr = ℏω a † a + βa †2 a 2 [17,18], with β a (typically small) dimensionless coefficient which is proportional to the refractive index shift induced by a single photon.In such a system, the energy to add an excitation is E n+1 − E n = ℏω(1 + 2βn), meaning that the deviation from harmonic behavior is linear in the intensity (proportional to photon number).A cavity with this third order nonlinearity would have its resonance frequency shift with intensity.Thus, the plots of Fig. 1c, for a photon in a Kerr medium, would be straight lines with slope 2β.This linear dependence arises from a low-order expansion of the nonlinear medium polarization in the cavity electric field: in the case of Kerr, the leading order nonlinear term would be third order, leading to a refractive index dependent on intensity.In contrast, for the case of DSC, the excitation energies do not vary linearly in photon number, but instead are very high-order near the critical photon number (as the function L n (4g 2 ) is practically exponential for n ∼ g 2 ): as if the cavity contained a nonlinear medium whose polarization had a non-perturbative dependence on intensity.
In what follows, we show how this non-perturbative nonlinearity arising from the spectrum of the Rabi model leads directly to the possibility of new effects, such as (1) systems which become reflective to light when a large but fixed number of excitations are in the cavity (N -photon blockade), and (2) lasers which generate intracavity Fock states in their steady states, as opposed to coherent states, as conventional lasers do.

II. COHERENT DRIVING AND N -PHOTON BLOCKADE IN A SYSTEM WITH DEEP-STRONG COUPLING
The nonlinearity perspective presented here, although not previously noted in the literature, is largely based on the known spectrum of DSC systems.We now use this perspective to develop the main new results of this paper.
First, we establish how these systems become reflective to light when a large, but fixed number of photons are present in the cavity.In other words, we demonstrate the fundamental phenomenon of N -photon blockade.To do so, we illustrate the dynamics of DSC photons under coherent driving by an external signal (e.g., an applied microwave signal, or an external laser, at frequency ω).Thus, to the Hamiltonian of Eq. ( 1), we add a driving term of the form H drive = η(X (+) e −iωpt + X (−) e iωpt ), where η and ω p are the strength and frequency of the drive.Additionally, X (±) are the positive and negative frequency components of the operator X = b + b † , where b is the annihilation operator of the DSC boson defined earlier.Specifically, the positive frequency component is defined in terms of X as X (+) = m<n X mn |m⟩ ⟨n| .The negative frequency operator is then In Fig. 2a, we show the probability distribution p(n) for the number of DSC photons n, as a function of the time t after the coherent drive is turned on.We note that n enumerates over states in the "down" manifold of spin states.For the parameters we use, the population which leaks out into the ther manifold is small.Fig. 2b shows slices of this probability distribution, and the corresponding Wigner functions for the DSC photon (for σ = −1).Immediately after the pump is turned on, the average excitation number begins to grow in a manner which is similar to that of a "normal" (i.e.linear) pumped cavity.This is expected, since for low excitation numbers, the DSC bosons have an almost perfectly harmonic spectrum.However, as soon as the probability distribution approaches the strongly anharmonic point N ∼ g 2 , the probability distribution begins to compress, as it becomes harder for photons to be added to the cavity.The driving frequency which was once resonant for lower photon numbers becomes highly non-resonant at the blockade point, resisting the population of excitations into any state beyond N .As the blockade point is approached, the quantum state of light deviates further from the classical coherent state which is produced with a linear resonance, as evidenced by the Wigner functions which take on a negative (blue) value with many fringes.
Once the blockade point is hit, the distribution actually turns around, and then repeats the cycle.This extreme form of the behavior occurs when the timescale associated with the field growth is faster than those associated with dissipation in the system.In the SI, we show the influence of dissipation.When this dissipation is present, the dynamics are similar for short times to what is shown in Fig. 2: after dissipation be- The mean photon number initially grows in accordance with a harmonic spectrum, but is abruptly stopped at the blockade number N due to the sudden anharmonicity in the energy spectrum.(b) Probability distribution slices and Wigner functions at selected times (shown by vertical lines in panel (a)).The probability distribution initially evolves as an approximate coherent state, but then acquires a reduced variance at the blockade point.Interference fringes in p(n) appear due to the nonlinear squeezing that occurs.After the blockade point, the distribution turns around as it is reflected due to the blockade.Parameters used are g = 5, λ = 0.1, η = 0.005, and ωp set to the difference between the two lowest energy eigenvalues in the "down" manifold of spin states.gins to act, the system reaches a steady state which can be squeezed in DSC excitation number, having a variance which is below the mean (steady-state squeezing tends to be fairly modest in this configuration, about 3 dB).

III. LIGHT EMISSION IN THE DEEP-STRONG COUPLING REGIME
Specifically, we study how light emission is modified by these photonic quasiparticles.Unlike most studies of light emission with photonic quasiparticles (reviewed for example in [15]), here we look at the unique modifications coming from the nonlinear properties.Consider an external qubit (denoted 'em', for emitter) coupled to this DSC photon.The exact form of the coupling depends on the circuit implementation.To keep the discussion concrete, we will consider a FIG.3: Fock lasing due to equilibrium between high-order nonlinearity and dissipation.(a) Light emission of DSC photons can be understood in terms of the coupling of an emitter (e.g., a probe qubit) weakly coupled to the DSC system, as might be realized by coupling a superconducting qubit to a flux-qubit-LC-resonator system.The probability to stimulatedly emit DSC photons scales as n + 1 for small n, and then sharply decreases due to the sudden anharmonicity for n > nc ∼ g 2 ."TLS" denotes two-level system.(b) This behavior leads to a gain medium whose gain coefficient (green lines) is highly nonlinear.The quantum state of DSC photons will depend on how this nonlinear gain comes into equilibrium with the loss (red lines).(c) Steady-state intensity and power fluctuations of lasers in different coupling regimes as a function of pump intensity.For the "harmonic" regimes (weak, and deep-strong), a rapid growth in intensity at threshold is seen.In contrast to the weak coupling regime (as in a "normal" laser; light blue curve), a laser operating in the deep-strong coupling regime has its intensity saturate, and its fluctuations vanish at high pump, converging to a high-number Fock state (dark blue and purple curves), leading to Fock-like statistics (right).(c, bottom) Statistics for different pump strengths for a single coupling, showing evolution from thermality to Fock-like statistics.
simple coupling Hamiltonian of the type which couples the emitter directly to the DSC photon.Regarding the assumed form of the Hamiltonian, we note that our conclusions are not particularly sensitive to the exact form of the interaction [43].What we do assume however is that ϵ is small, so that the coupling of the external emitter to the DSC system is weak (ϵ ≪ ω).Thus, the system in mind is a single resonator coupled to two qubits, one with weak coupling and one with deep strong coupling, as illustrated in Fig. 2a.
To understand emission and absorption of DSC photons, consider the case in which the qubit is in its excited state |e⟩ and there are n DSC photons present of spin −1 (e.g., occupying the state |n, −1⟩ of Eq. ( 2)).If the qubit is at frequency ω (same as in Eq. ( 2)), then the qubit transition will be nearly resonant with the transition n → n + 1 of the DSC photon, provided n ≲ n c .The dynamics can be restricted to the subspace {|e, n⟩, |g, n + 1⟩}, and the probability of (stimulated) emission P (n + 1) is simply given by Here, δ is the dimensionless detuning of the emitter and ω, such that ω em 0 − ω ≡ ωδ.Eq. ( 4) is the direct consequence of the Jaynes-Cummings dynamics of a two-level system (emitter) with a boson (DSC photon) with some detuning.The detuning depends on excitation number due to the nonlinearity of the DSC photon, and the detuning sharply rises near n c (Fig. 1c).In Fig. 2a, we plot the stimulated emission probability as a function of n after a small evolution time t ≪ ϵ −1 and for δ = 0.For n < n c , ∆ n+1 ≈ 0, that probability is simply (n + 1)(ϵt) 2 , corresponding to stimulated emission proportional to n + 1, as expected for conventional photons.For n ≳ n c , the emission probability drops rapidly, because of the corresponding rapid increase in ∆ n .This can be understood as a type of N -photon blockade, in which a system can readily accept N excitations, but not N + 1.For N = 1, this corresponds to the conventional photon blockade observed and discussed extensively in strong coupling cavity QED (in which the polariton anharmonicity is strong at the level of one photon).In the "conventional" strong-coupling cavity QED case, this N = 1 blockade corresponds to single-photon nonlin- A harmonic-to-anharmonic crossover occurs for a maximum photon number nmax ∼ g 2 for which the propagation function goes to zero.When this happens, the probability of having photons larger than nmax vanishes.Near threshold (where the effective temperature of the photon goes to infinity), this leads to nearly uniform states of the electromagnetic field sharply cutoff at the maximum photon number (right panel).(b) When the decay rate of the gain medium is large, the anharmonic region becomes narrower (bottom left), and for sufficient pump intensity, the photon distribution can "tunnel" through the barrier, evolving effectively as a coherent state.In this tunneling regime, the distribution becomes bimodal, taking on the characteristics of the Fock and coherent states for some pump parameters (bottom right).
earity which is highly desired for many applications.Here, the spectrum is such that the photon blockade is delayed to N = N c photons, leading to an exotic and strong quantum nonlinearity that operates at N ≫ 1 photons.

A. A new type of laser
Eq. ( 4) displays one of the main results: that the highorder nonlinearities arising from non-perturbative quantum electrodynamical coupling lead to a type of gain (stimulated emission) that is correspondingly non-perturbative in intensity.One may imagine that this type of nonlinear stimulated emission would have implications for lasers -or in this case, masers, given that the most imminent implementations, based on circuit QED, would be at microwave frequencies.We will stick to the term "laser" since it has largely subsumed masers.In this section, we show that the DSC-based gain discussed before creates lasing into high-order Fock states (rather than coherent states).
We now show how the nonlinear gain provided by the coupling of an excited two-level system to DSC photons can re-sult in a laser with new steady state photon statistics.To capture the resulting lasing dynamics in a quantum mechanical way, we shall find an equation of motion for the reduced density matrix ρ of the DSC photon (tracing out the gain medium).This equation takes into account both the stimulated emission dynamics and the loss dynamics associated with, for example, leakage from the cavity (which we take here for simplicity as the primary loss mechanism for the DSC photon).In the SI, we derive the equation using several methods, all in agreement with each other.Here, we focus on the equation for the DSC photon occupation probabilities, ρ nn .Assuming that excited states of the gain medium are pumped at rate r, the equation of motion for the DSC photon density matrix is found to be: Here, R n = 2rϵ 2 Γ 2 +Fn is the stimulated emission coefficient, with . The R n are plotted (green curves) in Fig. 2b for different coupling strengths.For weak coupling, it is simply saturable gain R(n) ∼ 1/(1 + n/n s ) with n s the saturation photon number.For DSC, we see that the gain coefficient is given by the standard saturable form for n < n c and then rapidly decays for n ≳ n c (with occasional oscillations arising from the oscillatory behavior of the Laguerre polynomials).Here, κ n = κ|⟨n − 1|a + a † |n⟩| 2 , with κ the decay rate of the cavity in the absence of DSC (see SI for derivation).We note that for simplicity, the gain medium has been taken to have population and coherence decay rates arising from the same source (so that Γ = 1/T 1 = 2/T 2 ).This simplifies the calculations but does not qualitatively change our conclusions.
The steady state photon probability distribution is entirely different from that of a traditional laser, which produces a dephased coherent state.To quantify this, we solve a recursion relation to obtain the steady-state probability distribu- In Fig. 2c, we show the intra-cavity photon number and photon fluctuations for DSC in comparison with weak coupling.We also present the corresponding photon statistics.In the weak coupling regime, the photon number as a function of pump follows the canonical "S-curve" relating the input pump and output intensity of a laser.The output intensity grows sharply for pump beyond the threshold pump level, r th = κΓ 2 /2ϵ 2 .The fluctuations below threshold are essentially those of a thermal state, and far above threshold, grow according to shot-noise (as √ n, as for a Poissonian distribution corresponding to a randomly-phased coherent state): this is the textbook result of the laser theory of Lamb and Scully [19,20].In contrast, the "Fock laser" (g = 5, 10, 18), saturates (at n c ∼ g 2 ), and the photon number fluctuations go to zero, leading to the quantum statistics of a Fock state (Fig. 2c, right) as the pump increases.Fig. 2c (bottom) further shows how the photon statistics evolve with pump and coupling (taken for g = 10; additional results shown in SI).Beyond threshold, the distribution of pho-tons (for DSC) approaches that of a thermal state of negative temperature.Such states, as the pump is increased (and T → 0 − ), approach states where only the highest-most level is filled, with minimal spread, which closely approximates a Fock state of n c DSC photons.
To understand this Fock lasing effect, it is helpful to refer to the gain and loss curves of Fig. 2b, as well as the steady-state The steady-state distribution has the property that the probabilities are maximized where gain equals loss, and probabilities are suppressed if one of gain or loss far exceeds the other.In particular, the larger the angle between the gain and loss curves (at the crossing point), the tighter the concentration of probabilities about the mean.Increasing the pump rate r further will scale the gain curve up, leading to a steeper slope and further suppression of photon number fluctuations, leading asymptotically to a Fock state.
Beyond these close approximations to high-photon-number Fock states, other unusual states can arise from the equilibrium between gain and loss, due to this sudden anharmonicity for n ≳ n c .For example, near threshold, where the number fluctuations increase dramatically, the resulting distribution is nearly step-like, going to zero rapidly for n c .This anharmonicity provides a "wall" for the photon probability distribution that is too hard to pass through, even as the fluctuations get very large near threshold.These effects also depend on the decay rate of the gain medium: if the decay rate is high, then it provides gain over a large bandwidth, and so changes in the DSC photon frequency have a reduced effect on the stimulated emission rate R n .As a result, for increasing pump, the distribution can "tunnel" through the wall, leading to states that interpolate between Fock and coherent states, as well as pure coherent states for large enough pump.

IV. DISCUSSION AND OUTLOOK
Comparison to other approaches The new type of infinite-order nonlinearity realized by the Rabi Hamiltonian in the deep-strong coupling regime, in principle, enables the deterministic creation of large Fock states, which has proven challenging in general.In this section, we briefly review other approaches to generating Fock states, especially those relevant at microwave frequencies.It is currently possible to deterministically produce Fock states of order roughly 15 in microwave resonators through a combination of external driving of the cavity by microwave pulses and superconducting transmon qubits [21].Fock states have also been generated in microwave cavities by strongly coupling them to transmon qubits that are repeatedly pumped to inject photons into the cavity at deterministic times [22].Older foundational work in the field of cavity quantum electrodynamics made use of Rydberg atoms strongly coupled to microwave cavities in order to generate low-order Fock states using principles such as the one above, as well as quantum feedback protocols [23][24][25].Such Rydberg atom-cavity interactions form the basis for new theoretical proposals to extend microwave Fock states to higher photon numbers [26,27].
Compared to these approaches, the approached outlined here has some notable advantages.The "Fock laser" shown here, which populates a cavity with N excitations based on stimulated emission, does so because of a strong anharmonicity of the spectrum.As a result, any approximately monochromatic pump of energy which incoherently pumps the cavity can lead to population of a Fock state.This is in contrast to approaches such as the "micromaser" in which Rydberg atoms, which interact with the cavity in the conventional strong-coupling regime, are injected into a cavity one-at-atime.There, Fock states can in principle based on the concept of "trap states" [23,24]: for a certain photon number n and interaction time between atom-and-cavity τ , the probability of adding a photon can vanish (due to a vanishing matrix element), leading to a fixed point where a Fock state of order n can be populated.This effect therefore is not robust against loss, multiple atoms being present at a time, inhomogeneities in the interaction time, etc.This is why it has proven difficult to produce good approximations to high-number Fock states in practice.
Compared to transient approaches such as in [21,22] approximate N -photon state is a steady-state solution, even in the presence of loss.As a result, the Fock state can be maintained for arbitrarily long, in contrast to "transient" approaches, in which the Fock state lasts only for the cavity lifetime.
Implementation: Of course, the primary limitation in our approach has to do with using deep-strong light-matter coupling, which requires extremely strong light-matter coupling strengths.That said, recent work on realizing deep-strong coupling of superconducting qubits to a microwave (LC) resonator, as in [9,14], provides a path to observing the effect predicted here.It is already possible to have control over g from weak coupling to a value of nearly 2. With a g of 2, one can see from Fig. 1 that a Fock state of three or four excitations could be pumped.For smaller g, in the ultra-strong coupling regime where 0.1 < g < 1, only one excitation can be created, as a manifestation of the conventional photon blockade effect [28,29].Thus, the behavior of our model from weak to (modest) deep strong coupling can already be realized.
Regarding the gain medium, it is important to point out that while a typically gain medium consisting of many emitters, the physics can also be realized by a gain medium consisting of a single qubit.The qubit should be weakly coupled to the same cavity as the strongly coupled qubit, and will lase, provided that the gain from this one qubit is above threshold [20].Single-qubit gain is responsible for much of the exciting experiments on "one-atom lasers" (in real [30,31] and artificial atoms [32][33][34]), in which a single atom or artificial atom provides enough gain to lase.
Thus, a conceptually simpler − and perhaps more attractive − approach to realize our predictions is to consider a gain medium consisting of a continuously pumped superconducting qubit which is weakly coupled to the same resonance as the strongly coupled qubit (which for example happens if ϵ ≪ κ).In Fig. 2, we took ϵ = 10 −5 ω, and Γ = 10 −3 ω.
Thus, for a single gain qubit, threshold is reached provided the quality factor of the resonator is above 5 × 10 6 .There are two advances that would support reaching larger g values: the rapidly increasing coupling constants that have been realized with superconducting qubits (see Fig. 1 of [8]), and early estimates in this field suggesting the possibility of g values of roughly 20 [35].Another important point is that while we have focused in this paper on incoherent pumping (based on emission from two-level systems), the nonlinear emission physics described in this manuscript could also be extended to coherent pumping of the DSC photon by an external microwave signal.In that case, we expect that by combining the high-order nonlinearity of the DSC photon with a frequencydependent leakage loss (e.g., loss coming from a reflection filter), one could engineer a highly nonlinear loss which would be "dual" to the highly nonlinear gain introduced in Fig. 2.
Summarizing, we have shown a physical principle -using non-perturbative photonic nonlinearity -which could enable lasers that produce deterministic, macroscopic quantum states of light, such as Fock states.Part of the new physics uncovered here, related to lasing in systems with sharply nonlinear gain, could in principle also be extended into the optical regime.In fact, in [36,37] − inspired by the developments in this manuscript − we discuss how trying to mimic the new "Fock lasers" predicted here, but at optical frequencies.This is done essentially by combining a highly frequencydependent loss with Kerr nonlinearities to get an effectively non-perturbatively nonlinear loss.Thus, the principles established here, independently of deep-strong coupling, should also give rise to new ideas and experiments in the optical domain.dom, quantum measurements, and time-dependent interactions in cavity and circuit qed.arXiv preprint arXiv:1912.08548,2019.
[43] The term σ em x (b+b † ) contains an interaction between the dipole moment of the emitter and that of the qubit.Such interactions are to be generically expected, as especially emphasized in recent works on superradiant phase transitions, as well as gauge invariance in ultrastrong coupling cavity and circuit quantum electrodynamics [38][39][40][41][42].We could write the term in question as ασ em x σx.Here, α = 2ϵg.Because this dipole-dipole term leads only to changes in spin quantum number, and not changes in excitation number (see SI), and because the spins are not separated by ω, these terms have little effect on the dynamics of the photon number probabilities that we consider.For example, we find that ignoring this term altogether leads to the same conclusions.Hence, for the purposes of the manuscript, we have taken a simple coupling that illustrates the physics best (emission of a "b" particle by an emitter).

FOCK LASING BASED ON DEEP-STRONG LIGHT-MATTER COUPLING
In this Supplement, we derive and extend the results of the main text.Consider a system involving matter coupled to a cavity mode very strongly, so that the system is in the ultra-or deepstrong coupling regime.This system is described by the Rabi Hamiltonian of Eq. ( 1) of the main text (Hamiltonian and variables re-defined here for self-containedness): Here, ω 0 is the transition frequency of the two-level system, σ x,z are the x and z Pauli matrices, ω is the cavity frequency, a ( †) is the cavity annhilation (creation) operator, and g is the Rabi frequency.
We also non-dimensionalize the coupling as g = g/ω.
Let us now transfer energy into this system by means of external emitters, treated as two-level systems of energy ω 0 .Let us assume the emitter is primarily interacting with the cavity (as it is too far for direct interactions with the dipole of the matter).Let us then take the full Hamiltonian describing the coupling of one emitter to the light-matter system as which couples the emitter directly to the DSC photon.We can also consider interactions solely between the emitter and the resonator field, replacing b → a.We consider this case as well, to show that the exact nature of the emitter-qubit dipole-dipole coupling does not qualitatively change our conclusions.
If the emitter is in the excited eigenstate |e⟩, and it is resonant with a transition of the Rabi Hamiltonian, the emitter can transfer energy to the light-matter system.Upon interaction with a second emitter, if the next transition of the Rabi model has nearly the same frequency, the system can get further excited.A key observation is that in the deep-strong coupling regime g ≫ ω, the eigenstates are approximately equally spaced, and the excitations are oscillator-like, quite similarly to the zero-coupling case.This should allow the possibility of reaching a very high excitation number in the presence of many emitters, based on stimulated emission of these oscillator modes (We will call them DSC photons).When the coupling is not infinite, as in a realistic case, the levels are no-longer fully equally spaced.This detuning is photon-number dependent, thus acting as a nonlinearity which may qualitatively change the steady-state of this type of laser.
To begin, we need to derive simple forms for the eigenstates of the Rabi Hamiltonian in the deep-strong coupling limit.Then, we will consider their coupling to external emitters, and write a coarse-gained equation of motion for the density matrix of the DSC bosons, and then solve it.

Eigenstates of the Rabi Hamiltonian
In what follows, we will take ω 0 = ω (resonant) and λ = 0.In later subsections, we will analytically and numerically consider the case of a finite λ, which is found to preserve our main findings.
In the deep-strong coupling regime, we can treat the matter term in the Rabi Hamiltonian as a perturbation to the remainder of the Hamiltonian.The remainder of the Hamiltonian (divided by ℏ), which we call H DSC is where g ≡ g/ω is a dimensionless measure of the coupling strength.Introducing the displacement operator D(gσ x ) = exp gσ x (a † − a) , where we've taken g real without loss of generality, we have where we've omitted the overall constant −ωg 2 .From here, we can easily see that the eigenstates of this Hamiltonian are of the form D † (±g) |±x, n⟩, where |x⟩ denotes the x-spin basis, and n is a Fock state.In other words, the eigenstates involve the spin being x-polarized (rather than z-polarization), and the photon being in a displaced Fock state (rather than just a Fock state). Clearly, Clearly then, in this limit, the eigenstates are evenly spaced, and doubly degenerate.In fact, it can be seen as a system of two non-interacting bosons ("DSC photons").Introducing b σ = a + gσ x we can write the Hamiltonian as It can also be easily seen that [b, b † ] = 1.
The degeneracy of the DSC Hamiltonian is split by the matter Hamiltonian.We can find the resulting eigenstates and eigenenergies using degenerate first-order perturbation theory.The "good" eigenbasis of the problem is where σ = ±1, and a displacement operator without an argument implies that the argument is g.
The energies of the resulting states are where D n = ⟨n| D 2 (g) |n⟩ = ⟨n| D(2g) |n⟩.These eigenstates and energies are sufficiently accurate, even for g = 2 or g = 3.
Evaluation of D n .
Let us evaluate the D n .To do so, we write: To proceed, insert a "complete" set of states using the over-completeness of the coherent states.
That leaves us with Using the rule for the overlap of two coherent states, we have where we have written things this way to emphasize that α and α * are independent variables.We can now write this as Then, we transform variables as α → α − z and α * → α * + z * to get This can be generated from simpler integrals by differentiation, as: Completing the square in the remaining integral gives Shifting variables as α → α + 2z and α → α − 2z * , and performing the final Gaussian integral, we have These are related to Laguerre polynomials.To see this, take the derivative with respect to z * .We will also use the notation z → x/2 and z * → y/2 for clarity.
From the Rodrigues formula for the Laguerre polynomials, we then have Given that the z = g, we have then that the the level splitting is given by Time-evolution of the coupled system With the approximate eigenstates of the Rabi Hamiltonian, we now want to understand the full dynamics of H.We will take advantage of the fact that for a laser, ϵ is small, and in particular, ϵ ≪ ω, so that the rotating-wave approximation is valid.In this system, the rotating wave approximation consists of only considering the dynamics within degenerate subspaces of the unperturbed Hamiltonian The eigenstates of the problem are |k⟩ |nσ⟩, where now k = 0, 1 denotes emitter states (ground is zero, excited is one).The energies of such states are (up to a shift) where we have taken ω 0 = (1 + δ)ω.From here on out, let us assume δ ≪ ω.In that case, it is easy to see that the following four states form our nearly degenerate subspace: We now need to understand the action of the interaction Hamiltonian V ≡ ϵσ x,em (b + b † ) on this subspace.First of all, For k = −k ′ , we have From these matrix elements, we see that: if the pseudo-spin (σ) is conserved, then a non-zero matrix element occurs only when the boson number changes by 1. Noting that b = a + gσ x , we can also readily describe interactions using a as ⟨n nn ′ , such that: when the pseudo-spin changes, non-zero matrix elements occur only when the boson number is conserved.When the spin is conserved, the matrix elements are the same as for b + b † .Since only states with different photon number differ appreciably in frequency (and in particular, will be resonant with the emitter we introduce), the interactions are effectively the same whether we describe a or b.This is also to say that any modification in the coefficient of the dipole-dipole interaction between emitter and qubit will lead to the same result insofar as DSC photon dynamics are concerned.In Fig. S1, we show the matrix elements of a and b between adjacent states of the same spin, as well as a † a and b † b.For n < n c ∼ g 2 , they behave as one might expect for an oscillator.
We should note that beyond n c , these states and matrix elements that we calculate based on degenerate perturbation theory are expected to change significantly.However, the approximate result turns out to describe the system well because the probabilities to find photon numbers beyond n c are strongly suppressed in the Fock laser, rendering the description relatively insensitive to these details.Based on these considerations, we see that the Hamiltonian in the degenerate subspace may be written as: As we can see here, there are two independent blocks of the Hamiltonian (for each pseudo-spin) and we can thus study them separately.Let us assume that we're at zero temperature, and so the ground state has the − pseudo-spin, which we assume to be conserved for all times.In that case, we can work with the simple 2x2 Hamiltonian where the pseudo-spin label has been dropped.This can be written in terms of Pauli matrices as Introducing , we have very simply As we will see in the next section, we need to know how states of the form |1, n − 1⟩ evolve over time.Thus we need with Ûn = (∆n,0,ϵ √ n) √ Therefore So, the probability of remaining in the same state is while the probability of transitioning is

EQUATION OF MOTION FOR DSC PHOTONS
Now we consider the description of laser action.To do so, we formulate an equation for how the density matrix of the DSC photon changes due to stimulated emission by the emitter.The method of analysis presented closely follows the coarse-grained density matrix technique used to describe conventional lasers.It is described in many books, such as [? ?].Suppose we have our emitter coupled to the light-matter (DSC) system.The emitter unit starts in the state |i⟩ and the DSC system is taken to have a density matrix ρ DSC , so that the initial density matrix of the total system ρ tot is given by ρ tot (t) = |i⟩⟨i|ρ DSC (t).Let us look for an equation describing only the evolution of the DSC system.Assuming the interaction over a time T corresponds to the evolution operator U , we have that Let us express all operators in terms of their matrix elements, writing the above equation as where ρ DSC,nn ′ denotes the matrix elements of the DSC system, for a fixed pseudo-spin.More compactly, The DSC photon density matrix, ρ DSC = tr em ρ tot , can then be expressed as From here, a number of approaches can be followed.If there is no loss in the system, then the density matrix of the total system upon the next iteration is simply ρ DSC (t + T ) = |i⟩⟨i|ρ DSC (t + T ) and this procedure can be iterated in a discrete fashion.The evolution can also be seen as continuous if, over time T , the change in the density matrix is small.This doesn't describe the early stages of the evolution, but it can describe later stages once there are many bosons in the system.If there is a steady state, then the continuous evolution must describe the run-up to the steady state, as changes get smaller over time.In such a case, we have where r = N/T is the number of excited emitters introduced into the system in time T .We have also dropped the "DSC" subscript for the DSC photon for brevity.These terms in the evolution of the density matrix describe the gain in the system.In addition, since there are losses associated with the cavity, the emitter, and the matter coupled to the cavity, we need to describe those.For simplicity, we will assume the emitter has loss, and so does the DSC photon, but not the matter (qualitatively similar results arise if the matter has loss).

Lindblad terms
Here we describe the effect of dissipation of the DSC photon on the equation of motion for its density matrix.Let's assume for simplicity that the cavity loss the primary source of dissipation in the problem.For weak coupling, the standard prescription is to add a Lindblad term to the Liouvillian which prescribes the evolution of the density matrix.The Lindblad term would be As is well known from studies of dissipation in ultra-strong coupling of light and matter, the use of the standard Lindblad term leads to unphysical excitations (in the energy eigenbasis), even at zero temperature, and zero pumping [? ].Part of the issue is that in the USC regime, the a operator can create excitations in the eigenbasis, clearly not representing dissipation.Framed in terms of the standard derivation, the issue could be said that the interaction picture a operator has negative frequencies, and the use of white noise (with frequencies −∞ to ∞) introduces contributions from these negative frequencies [? ?].The issue can be rectified by keeping in mind the positive-frequency nature of the reservoir.
We now use this procedure to describe dissipation in the deep-strong coupling regime.Although the technique has been worked out for ultra-strong coupling, there is a commonly used assumption in the final result that all transitions have different frequencies, which does not necessarily hold in DSC, when the energy ladder is quasi-harmonic.Interestingly, as we will show from a physical dissipator, the issues described above create much less error in the DSC regime, and the use of an operator like a or b produces a similar result to a proper positive-frequency jump operator, as their negative frequency parts get exponentially suppressed.
Let us consider the Lindblad term arising from a system-bath coupling of the form where J is a DSC system operator (e.g., a+a † or b+b † ), and the b k are the bath operators, satisfying The couplings V k between system and bath are weak.To isolate the positivefrequency parts of J, we express it in its energy eigenbasis as J = n>m J mn T mn + m>n J mn T mn + n J nn T nn ≡ J (+) + J (−) + J 0 , with J mn = ⟨m|J|n⟩ and T mn = |m⟩⟨n|.
In what follows, we will consider the bath to be concentrated around ω, but broadband enough that the white-noise approximation may be made for any transitions we consider.For example, a bath with a half-bandwidth of 10% of ω would be sufficient for the values of g, V k we consider.
It would include all active transitions of the form n → n + 1, but would not include higher transitions (though the matrix elements for them are small anyway), and in the presence of a λ term, it would also not include transitions that only change spin (for λ = 0, the two spins are very nearly degenerate and so the argument should be treated with more care).Therefore, we may describe the interaction of Eq. ( 39) within the rotating wave approximation, instead considering We note that it is not necessary to take the RWA at this stage, but it makes the subsequent manipulations simpler.
Thus, we may approximate the evolution of the reduced density matrix of the DSC system (in the interaction picture) to second-order in time-dependent perturbation theory, as: where ρ I is the system-bath density matrix, ρ DSC,I is the system density matrix, V I is the systembath coupling in the interaction picture, and tr b denotes the partial trace with respect to the bath.
For simplicity, we will consider the bath at zero temperature.Upon taking the trace with respect to the bath, the term which is linear in V I will vanish, and the equation of motion becomes The first term may be simplified, taking the trace with respect to the bath variables, as where D(ω) is the density of bath states, and we have replaced the sum over k by an integral over bath frequencies.Since an operator of the form J (+) is a pure de-excitation operator, no spurious excitations are introduced, and the integration limits may be extended to −∞.Doing so, and making the white noise approximation, one immediately finds that the term evaluates to Let us use this to find the contribution of dissipation to the equation of motion for the populations, ρ nn .From here on out, we will suppress the "DSC" subscript.We will ignore the spin degree of freedom (and restrict the dynamics to a single spin ladder).Although this is not rigorous, one expects this to capture well the dynamics of the DSC photon number as, for λ = 0, one will just expect the nearly degenerate spins to be mixed, with little change of the oscillator quantum numbers.We validate this numerically.For finite λ the spin ladders can be split appreciably, and so they will decouple.Consider a J of the form b + b † .As discussed in the main text, b is a pure de-excitation operator, and b † is a pure creation operator.Therefore, J (+) = b.Using the fact that which is similar to the form one would expect for damping of a conventional photon.This is perhaps unsurprising in light of the fact that the DSC photon is essentially harmonic up to n c ∼ g 2 .
It is worth noting that the matrix elements derived for a, b in Eq. ( 24) are based on first-order degenerate perturbation theory.Beyond n c , these approximations do not hold up and the states and matrix elements change significantly.However, the approximate result turns out to describe the system well because the probabilities to find photon numbers beyond n c are strongly suppressed.
It is also worth noting that if we chose a instead of b as the jump operator, when we neglect spin, the matrix elements are the same.Numerically, we find that whether we choose a or b as the jump operator, negligible levels of excitations are created in the ground state, and the steady-state of the Fock laser we describe is not qualitatively changed.These numerics are shown in the last section.

Rate equations
We will now obtain a closed set of equations for the diagonals of the DSC density matrix, to get the probability of different Fock state occupations of the DSC photons.Setting m = m ′ , we have The set of equations for the coarse grained density matrix is only closed when U km,in (T )U * km,in ′ (T ) is zero unless n = n ′ .In that case, we have We can now note the conditions under which the equations for the populations become closed.We require U km,in (T )U * km,in ′ (T ) is zero unless n = n ′ .This is equivalent to saying that a transition in → km and in ′ → km are not simultaneously possible.Supposing i is also an eigenstate of the light-matter system, and that we are in the RWA, this statement appears to amount to energy conservation, as transitions are assumed to be only efficient if they are resonant, so that which, for a single oscillator, requires n = n ′ .
In the weak coupling regime then, we have (adding in the photon losses) Let's now consider the case of the emitter coupled to our light-matter system.Since we inject emitters in the excited state, we have i = 1.The state 1n couples only to 1n and 0(n + 1).So, the sum over probabilities leaves only the scattering matrix coefficients U 1m,1m and U 0m,1(m−1) .
Therefore, the coarse-grained equation simplifies to: We found these probability coefficients when studying the dynamics of the Hamiltonian in the degenerate subspace of fixed pseudo-spin.Plugging in the results there, we have [1] To proceed, we must the emitter loss (T 1 and T 2 decay) into account.Assuming that the emitter loss manifests as exponential decay with rate Γ, the effect is to average the probability coefficients over T with probability distribution P (T ) = Γe −ΓT .Noting that we have Noting that U 2 n = ∆ 2 n + nϵ 2 , we have Assuming resonance between the emitter and the light-matter system, we have finally with the nonlinearity, F (n) defined as Here, we have used For weak coupling, the statistics evolve from thermal to coherent with increasing pump.For the largest couplings considered, the state evolves from thermal (for low pump) to coherent (for intermediate pump) to a thermal state of negative temperature for higher pump.As the pump increases, the negative temperature state converges effectively to a Fock state.Note that the bottom left panel overlaps with Fig. 2 of the main text.
Our main results of nonperbative nonlinearity, as well as the Fock lasing action in these systems, are robust to the addition of this term.We summarize the main changes here.
We assume that the generalized Rabi Hamiltonian now takes the full form The spectrum is now approximately given by: where the mixing angle θ is defined by tan(θ) = ωD n /λ.
In principles, the modifications to the analysis of the laser action should follow through new additions to the matrix elements which couple these eigenstates.Specifically, we have However, we see that the only new terms are only nonzero when the photon number stays the same.Thus, the only modifications to the equations of motion come from the eigen-energies.This means that the equations of motion derived previously still hold valid, but with a new nonlinearity: Direct method for evolving the density matrix In the previous section, we treated the emitter-field interaction as if excited two-level systems were being injected into the system at rate r.We also treated the interaction with the emitters as sequential: as if one emitter interacts with the field at any given time, with probability coefficients averaged over the emitter's exponential decay probability.In this section, we provide an alternative treatment of the problem in which we consider the direct evolution of the density matrix in the presence of coherent emitter-field interaction, emitter pumping, emitter decay, and field leakage.
This approach, besides being in principle more rigorous, and besides providing further corroboration of our results above, also allows us to consider multi-level emitter systems, such as three-and four-level systems, which are more practical from the standpoint of lasers.This method has been applied to describe conventional lasers (see [? ]), but due to its generality, can be used to describe the Fock laser discussed in this paper.
The equation of motion for the density matrix is where is the Hamiltonian describing N multi-level emitters (with Hamiltonian H em,i ) coupled to the electromagnetic field associated with matter strongly coupled to a single electromagnetic field mode with coupling constant ϵ i .The levels a and b of the ith emitter are coupled to the field and comprise due to decoherence.Numerically, for this laser system, based on deep strong light-matter coupling, we also found that the steady-state (found by the null eigenvector of the Liouvillian (S such that ρ = Sρ)) is diagonal.Let us thus focus on the steady-state equations for the "photon diagonals" (n = n ′ ), which are simply where we have defined ∆ n+1 = ω an − ω bn+1 .
For simplicity, let us take γ a = γ b = Γ, so that rρ gn = Γ(ρ an,an + ρ bn,bn ) = Γ(ρ nn − ρ cn,cn − ρ dn,dn − ρ gn,gn ), where we have defined the photon populations ρ nn = (tr em ρ) nn in order to express everything in terms of these populations and arrive at a coarse-grained density matrix for the field.Let us now consider the case where γ c ≫ γ a and γ d ≫ γ b .In this case, we immediately see that ρ cn,cn ≈ 0 and ρ dn,dn ≈ 0. This is to say that these levels are depleted immediately after they are populated by the lasing levels.In this case, ρ gn,gn = Γ (r+Γ) ρ nn .The steady-state equations then reduce to the simple inhomogeneous equation: softens the anharmonicity (which can be understood from the term λ 2 + ω 2 e −4g 2 L 2 n (4g 2 ) in Eq. ( 60)).
The results are also insensitive to the exact form of interaction and dissipator (provided that the dissipator doesn't create spurious excitations).In Fig. S5, we show the steady-state, computed using interaction terms based on a or b, as well as dissipators based on a or b.

COHERENT PUMPING OF SHARP DSC NONLINEARITY IN THE PRESENCE OF LOSS
In the main text we showed how coherently pumped DSC bosons can show a striking effect of N -photon blockade due to sharp anharmonicity, generating highly nonclassical states of light.
For simplicity of discussion, the main text results did not include the effects of loss.We report on the effects of loss here, which were studied through solving the master equation associated with coherently pumped DSC bosons in the presence of damping.
To do so, we consider the equations of motion associated with a coherently pumped nonlinear oscillator.We assume that the driven nonlinear oscillator has a Hamiltonian where η is the drive strength, and ω is the drive frequency.Additionally, ω n is the spectrum of the anharmonic oscillator, which for the DSC system is computed using the methods described in earlier sections of the paper.In a frame which rotates with the frequency of the drive, and after making the rotating wave approximation, the pumped system has the time independent Hamiltonian Here, δ n is a detuning from the pump frequency defined as ω n = nω + δ n .In this sense, δ n encodes the anharmonicity of the spectrum with respect to the coherent driving frequency.The time evolution in this rotating picture is given by the density matrix equation Here, L[ρ] is the Liouvillian operator acting on the density matrix.Additionally, L[ρ] is a standard Lindblad term which performs linear damping on the a operator with a rate κ.From this, we can compute time evolution of the states, or solve the steady state condition Although the Hamiltonian is not exactly the same as the driven Hamiltonian presented in the main text in the absence of loss, we find that the models behave nearly identically in the limit of low loss, indicating that the simplified model presented here provides a good description of the key physics.The main advantage of this approach is that in the rotating frame, the Hamiltonian becomes time independent, enabling numerical solutions of the steady state solution that do not require time evolution of the density matrix.We can also see how these states are created transiently.Fig. 6d shows how the representative sub-Poissonian (green) state is created when pumped from the ground state, in terms of its probability distribution.In the early stages of time evolution, the mean photon number grows quadratically in time, as it would for an ordinary harmonic oscillator.This occurs since at early times, the pumped field only occupies the parts of the DSC boson spectrum which are essentially perfectly harmonic.However, once the tail of the probability distribution approaches the blockade number N , nonlinear oscillations begin to occur, as the state is reflected from the boundary.Eventually, these oscillations damp out, and a steady distribution is reached, corresponding to the green curve in Fig. 6c.We observe that in the presence of losses, it appears that the minimum attainable Fano factor is around 0.5.
Fig. 6e shows how the mean and variance evolve as a function of time for the states shown in Fig. 6c.We see that the lowest pump level generates a coherent state in a manner nearly identical to a linear cavity.The subpoissonian state is generated with influence from the blockade, as described above, and a few cycles of nonlinear oscillation are seen in the mean and variance as the steady state is approached.Finally, the strongest coherent pump results in a super-Poissonian distribution, which is reached after rapid nonlinear oscillations.We note that even though the mean photon number is only marginally higher than that of the sub-Poissonian example state, the variance is higher by more than a factor of 3.
We finally comment on the key differences between these results and the lossless calculations shows in the main text.One key difference is that the lossless calculations in the main text exhibit a periodic "revival" behavior as the probability distribution oscillates between the ground state and the blockade number.Instead of these huge oscillations, the damped system exhibits oscillations as it finally settles into a steady state.Relatedly, the presence of loss eliminates the presence of fringes in the probability distribution as the state hits the blockade number and squeezes.In this sense, the loss has the effect of moderating the extent of quantum features which can be realized in the state.This is entirely consistent with the well-known fact that dissipation acts as a form of decoherence which degrades quantum states.Nevertheless, even in the presence of loss, this coherently pumped system exhibits clear features of N -photon blockade, and can be used to generate DSC boson quantum states with sub-Poissonian character.This coincides exactly with the equation of motion of the so-called micromaser, which describes the interaction of injected two-level atoms interacting with a cavity (in the perturbative coupling regime g ≪ ω).This is quite interesting as the micromaser equations assume g ≪ 1, while here, we are starting from the limit g ≫ 1.Moreover, by averaging over decay times as we do in the next subsection, we will find exactly the standard Scully-Lamb master equation for a conventional laser.What's happening here is that in the weak-coupling regime, assuming the emitter is resonant, the detunings also approximately vanish between the nearly degenerate levels.And so we get a similar equation, except that it the conventional case, it is in the photon basis, and here it is in the DSC photon basis.

FIG. 2 :
FIG. 2: Coherent pumping of DSC bosons leads to N -photon blockade.(a) Probability distribution p(n) of DSC photon number n as a function of time t in the presence of a coherent driving field.The mean photon number initially grows in accordance with a harmonic spectrum, but is abruptly stopped at the blockade number N due to the sudden anharmonicity in the energy spectrum.(b) Probability distribution slices and Wigner functions at selected times (shown by vertical lines in panel (a)).The probability distribution initially evolves as an approximate coherent state, but then acquires a reduced variance at the blockade point.Interference fringes in p(n) appear due to the nonlinear squeezing that occurs.After the blockade point, the distribution turns around as it is reflected due to the blockade.Parameters used are g = 5, λ = 0.1, η = 0.005, and ωp set to the difference between the two lowest energy eigenvalues in the "down" manifold of spin states.
Photon number 0.0

FIG. 1 :
FIG. 1: Matrix elements of a † a, b † b and a, b, showing that b counts excitations of the DSC system over the full range of eigenstates.However, up to n c , a and b act similarly.

I 2 J
(t)ρ DSC (t), where κ = 2πρ|V | 2 .A similar manipulation for the remaining terms yields that the free dissipation dynamics of the DSC Hamiltonian are governed by ρDSC,I = − κ

∆n = 1 FIG. 3 :FIG. 4 :FIG. 5 :
FIG.3: Steady-state of the Fock laser, calculated numerically, by finding the steady-state of the Liouvillian operator.Plot shows the unpolarized probability distribution for λ = 0.For ϵ = 10 −5 ω, κ = 10 −8 ω, and r = 10Γ (such that the population inversion of the gain is about 90%, the resulting state is nearly a Fock state of 100 DSC photons, with a residual uncertainty of 1.This state has noise 99% below the shot noise level.Moreover, this calculation shows that the Hamiltonian of Eq. (3), coupled to damping, supports Fock states as its steady state, from first principles.

Figure 6 FIG. 6 :
Figure6shows both steady state and transient behaviors for DSC bosons pumped with coherent drive strength η and loss rate κ.Evidence of the blockade effect can be seen in Fig.6a, where