Engineering first-order quantum phase transitions for weak signal detection

Detection of weak quantum signals is central to precision metrology,^1,2 search for new fundamental particles,³ studying gravitational effects,⁴ and quantum information science.^5,6 An early example of a weak signal detector is the bubble chamber for charged particle detection in high-energy physics experiments.⁷ Here, superheated liquid vaporizes into gas bubbles, which traces out the path of the charged particle. Another more recent example of a quantum detector is the superconducting nanowire single-photon detector (SNSPD).⁸ Here, a single photon causes a click event in a critically biased superconductor by inducing a phase change to the normal conducting state. The common theme between these two detection mechanisms is a weak signal inducing a thermodynamic phase transition. This concept recently inspired a new class of detectors that function by a weak signal triggering a quantum phase transition (QPT). To emphasize the need for a critical biasing field and a quantum (as opposed to classical) phase transition, we address this device as the quantum critical detector (QCD).⁹ 

The word “critical” in QCDs is used to emphasize the need for a critical biasing signal, which is not required in widely utilized quantum linear amplifiers. The philosophy of operation is similar to a conventional single-photon triggered avalanche process where a weak signal (i.e., single-photon pulse) triggers an avalanche in an optimally biased system leading to a large number of electron hole pairs. The key difference is the readout mechanism (amplification) in the QCD, which is not an avalanche but is related to the phase transition that causes a macroscopic excitation in a single bosonic output mode or an abrupt change in the long-range order of spins. The concept of a “click” or single-shot quantum detection event is related to the nonanalyticity that occurs near the phase transition point as a parameter in the detector is perturbed. QCDs necessarily require a first-order phase transition to ensure a giant change in macroscopic observables arising from a weak perturbing signal. Our focus is not the critical point of continuous phase transitions, which have been recently exploited in quantum critical metrology.¹⁰ 

This approach of exploiting first-order QPTs to detect signals in a single shot is fundamentally different from repeated measurement schemes, parameter estimation, or quantum sensing/metrology.^1,2,11 The readout mechanism in these established schemes is related to interferometric processes, which can be enhanced through engineering quantum states of light^12,13 or matter.¹⁴ In stark contrast, the QCD does not utilize an interferometric readout but requires a quantum bias that prepares the detector in a predetermined phase close to the phase transition point.⁹ 

We also emphasize that the mechanism of quantum critical amplification is fundamentally different from the well established concept of quantum linear amplifiers.^15,16 In this traditional amplification approach, the weak input signal is directly amplified to generate a large output signal. The information carriers in input and output signals are usually of the same kind (e.g., bosonic excitations), and the gain of the amplifier is defined as the output-to-input ratio of signal amplitudes not particle number. The canonical example is the quantum linear amplifier, including the phase-preserving (phase insensitive)¹⁷ and parametric (phase-sensitive) amplifiers.^18–21 In his seminal work,¹⁶ Caves presented a comprehensive review of the fundamental quantum limit for linear amplification under this scheme. This quantum limit lays a lower bound on the minimum amount of noise added by a high-gain bosonic quadrature amplifier during amplification,^22,23 which has also been generalized to fermionic amplifiers.²⁴ 

In stark contrast, for the phase transition amplification scheme, the weak input signal functions as a control of an optimally biased phase transition system, which is significantly different from the quantum linear amplifiers. In these critically biased amplifiers, the input and output information carriers can be fundamentally different (e.g., input photons and output electrons), and the corresponding gain (i.e., the amplification factor) is defined as the ratio of the outputs with and without the input control signal. The classical critical detectors have been extensively used in practical experiments, such as the SNSPD,⁸ the single-photon avalanche diode (SPAD),²⁵ etc. The key amplification mechanism in the superconducting detectors is based on the thermodynamic (classical) phase transition triggered by the weak input signal. The QCD falls under this critical amplification scheme, and it is an open question whether quantum limits can be placed on this class of critically biased amplifiers.²⁶ 

In this paper, we show universal detection and amplification behavior in a class of models exhibiting quantum phase transitions. We use this to explain the amplification mechanism of a QCD. The detector model we introduce is closely related to the Dicke model²⁷ and the Lipkin-Meshkov-Glick (LMG) model.^28–30 We also introduce two new magnetic order parameters (OPs) to rigorously characterize the first-order QPTs in the detector model. By employing a mean-field theory as well as full numerics, we generate the complete phase diagram of the detector.

We uncover the first-order QPT in the LMG model, which is essential for our proposed QCD and has not been revealed in the previous literature. We find that the first-order QPT is fundamentally tied to the competition between two ferromagnetic phases with a long-range spin order in the $x$- and $y$-axes, respectively. We also predict that a universal first-order QPT exists in an interacting spin system with competing ferromagnetic orders.

We also numerically show that, at the first-order phase transition point of the detector, the sensitivity function $χ$ diverges with $N2$-scaling, where $N$ is the spin number. This scaling is much faster than previous first-order phase transitions^31,32 and provides extraordinary high sensitivity for weak signal detection. To understand the microscopic mechanism of the QPTs, we display the fundamental changes in the ground-state wave function during the phase transitions using Husimi $Q$-functions.

For natural atomic systems, the superradiant QPT in the Dicke model has been ruled out by the no-go theorem.^33,34 Here, we show explicitly how to overcome this no-go theorem. We show that the spin-spin (atom-atom) interaction can decrease the strong atom-field coupling required by the superradiant QPT significantly. This allows the superradiant QPT to occur.

Our work overcomes challenges in the simulation of quantum phase transition dynamics with weak signal perturbation. The dynamical evolution of a system near the phase transition point necessarily involves excited states which can show significant deviations from the conventional ground-state to ground-state transition behavior. Via direct time-dynamic numerical evaluation, we overcome this issue. We show the linear scaling in both the maximum quantum gain and the corresponding signal-to-quantum noise ratio (SQNR), which reveal high figures of merit of our QCD. We also use the time-dependent $Q$-functions to show the macroscopic changes in the bosonic output mode and the long-range spin order during the dynamical critical amplification in our QCD. Usually, the enhanced decay of the Loschmidit echo around the phase transition point is utilized to measure QPTs.^35,36 However, this enhanced echo decay exists in both first-order and second-order QPTs and only describes the deviation in the wave function from the initial state during the QPT. The enhanced quantum gain around the phase transition point addressed in this paper is a unique and universal characteristic of first-order QPTs, which captures the fundamental change in the macroscopic order of the detector.

Our proposed device can be obtained by engineering a multiqubit system to exhibit an artificial phase transition. We note that the simulation of quantum phase transitions has become a major recent area of interest. Second-order Ising-like QPTs have been demonstrated in experiments with trapped ions,³⁷ cold atoms,³⁸ and circuit QEDs.³⁹ As QPTs occur at zero temperature, our proposed QCD may have a higher signal-to-noise ratio and a lower dark counting rate than detectors utilizing thermodynamic phase transitions.

We also note that the detection of the axion, which is the prominent dark-matter candidate, is based on measuring single microwave photons generated from the axion-photon conversion process.^3,40,41 Currently, the efficient detection of propagating microwave single-photon pulses remains challenging due to the extremely low energy carried by the pulse.^42–44 Our proposed quantum signal detector based on first-order quantum phase transitions might pave a new path for microwave-photon counting and axion detection.

This paper is organized as follows. We first provide a general introduction to the QCD paradigm in Sec. II. Then, we introduce the Dicke-LMG model as an explicit detector model to demonstrate the QCD in Sec. III. In Sec. IV, we establish two new magnetic order parameters (OPs) to characterize the quantum phases of the detector and present the complete phase diagram of the detector obtained from the mean field theory. By splitting the Dicke-LMG model into three submodels, we study the first-order and second-order QPTs existing in the full model and the fundamental changes in the ground-state wave function during the phase transitions in Sec. V. In Sec. VI, we demonstrate the dynamical quantum critical amplification in the QCD by exploiting the giant sensitivity of the first-order QPT. In Sec. VII, we list some possible platforms to demonstrate our QCD. Finally, we collect our main conclusions in Sec. VIII. The details of the mean-field theory are given in  Appendix A, and the numerical approach used in this paper is presented in  Appendix B.

In our previous work,⁹ we proposed a prototype QCD by exploiting the giant sensitivity in a first-order QPT. Our QCD is inspired by the routinely used classical critical detectors, such as the SNSPD^6,8 and the bubble chamber.⁷ The amplification of these classical detectors is based on the ultrahigh sensitivity of the detector at the phase transition point of the thermodynamic (classical) phase transitions. Our QCD is the first quantum analog of the classical critical detectors. In this section, we explain the input signal, the amplification mechanism, and the macroscopic output signal of our QCD explicitly. We also explain why first-order QPTs are essential for quantum critical amplification.

To show the analogy between our proposed QCD with the conventional detectors, we deconstruct the SNSPD to explain the amplification scheme in classical critical detectors. The input of the SNSPD is a single-photon pulse—an extremely weak quantum signal. The core element of the SNSPD is a superconducting nanowire with a typical width of $100$ nm, a thickness of $8$ nm, and a length of $10μ$m.^46,47 The current in the superconducting nanowire is biased very close to the critical current; thus, even a single-photon pulse can break the superconductivity.⁴⁸ The output signal is the voltage difference between the two ends of the superconducting nanowire. In the transduction (absorption) process, the incident single-photon pulse generates one resonantly excited electron. As the center frequency of the pulse is much larger than the energy gap of the superconductor, this highly excited electron will break hundreds of Cooper pairs via the strong electron-electron interaction. Then, the local temperature around the excited electron increases to form a hot spot. This hot spot diffuses and reduces the local density of the superconducting electrons within this small hot region. Finally, a phase transition from a superconductor to a normal metal in this small cross-section slab occurs and blocks the superconducting current in the nanowire. An observable output voltage pulse is generated to realize the amplification. Here, we see that the high sensitivity in the superconducting phase transition and the critical bias current play the key roles in weak quantum signal amplification.

Next, we show the analogy between a QCD and the SNSPD and explain the macroscopic output signal of our proposed detector.

The full measurement in our QCD is split into two main processes: transduction (absorption) and amplification. In our proposed QCD, absorption of the incident weak signal leads to a small time-dependent variation in a relevant parameter of the detector system instead of the temperature. The amplification is realized by the QPT triggered by this parameter variation, in contrast to the thermodynamic phase transition triggered by the temperature change in an SNSPD.

The input signal of our QCD can be an arbitrary quantum weak signal, such as a single-photon pulse. After absorption of this input signal, a small time-dependent variation in the detector parameter is generated. Similar to the classical critical detectors, we also need to bias our QCD very close to the phase transition point to guarantee that this small parameter variation can cross the phase boundary to trigger a QPT. As explained later, when a first-order QPT occurs, a large output change can be obtained to complete the amplification. We see that the amplification scheme in the QCD is the same as the classical critical detector by only replacing the thermodynamic phase transition with a QPT.

In practice, we can engineer and select which parameter of the detector to be changed by the input signal. One example is the interaction strength within the detector, e.g., the spin-boson coupling or spin-spin coupling shown in this paper. It can also be a small effective magnetic field change coming from the magnetic dipole of the absorber, such as a nitrogen-vacancy center.⁴⁹ The dynamics of the absorption of a quantum pulse can be theoretically incorporated into the dynamical quantum critical amplification.^50,51 Without loss of generality, we model the transduction (absorption) process as a temporal change in the detector parameter, which is assumed to be proportional to the input signal absorption probability for simplicity. For a specific realization of the QCD, the transduction process can be numerically simulated via the quantum pulse scattering theory.^51,52

The output amplified signal of a QCD is the macroscopic change in one of the OPs. One simple example demonstrated in this paper is the superradiant OP, i.e., the macroscopic excitation in a single bosonic mode. Before the absorption of the incident weak signal, the detector is biased in the ground state of the phase, in which the bosonic mode is in the vacuum state. Finally, the bosonic mode evolves to a state with macroscopic excitations after the first-order QPT is triggered by the input signal. The macroscopic population in the bosonic mode can be readout via a classical device directly.

The detector model as well as the corresponding readout channel presented in this paper is only one explicit example to show the amplification mechanism of the QCD. In practice, we can engineer the QPTs, readout channel, especially the interaction between the detector and the input signal to detect different kinds of particles, such as photons, charged particles, axions, etc. Another output signal of the prototype QCD in this paper could be the magnetic noise change, i.e., read out the in-plane magnetic fluctuations of spins with spin noise spectroscopy.⁵³ 

Now, we explain why first-order QPTs are essential for QCDs. A QPT describes an abrupt change in the ground state of a many-body system at zero temperature.⁴⁵ Most of the QPTs discovered in physical systems are of second-order, like the QPTs in the Ising model,⁵⁴ the Hubbard model,^55,56 the Bose-Hubbard model,⁵⁷ the LMG model,^28–30 the Dicke model,^27,58–60 etc. Here, we emphasize that first-order instead of second-order QPTs are required for quantum critical amplification. We compare the differences between first- and second-order QPTs schematically in Fig. 1. From panel (a), we see that the order parameter in second-order (or higher-order) QPTs changes continuously at the phase transition point $λc$⁠. No macroscopic change in the OPs, which functions as the output signal of a QCD, exists during a second-order QPT. Even at the phase transition point, a large parameter variation is required to obtain an observable change in the system. This large parameter variation cannot be induced by a weak input signal, such as a single-photon pulse. Thus, a high quantum gain cannot be obtained using second-order QPTs, which limits their practical applicability for weak signal amplification. In contrast to the continuous phase transitions, the order parameter changes discontinuously at the phase transition point in a first-order QPT. Thus, even a very small parameter change at the phase transition can lead to a significantly large change in the values of the OP (the output signal). This elucidates the great potential of first-order QPTs for high-gain amplifiers.

To characterize the sensitivity of a phase transition system, we defined a sensitivity function as the first derivative of the order parameter. From panel (b), we see that a nonanalytical “kink” exists in the sensitivity function of both first- and second-order QPTs. However, the height of the sensitivity function at the first-order QPT point, which diverges in the thermodynamic limit, is much higher than that of a second-order QPT with a finite peak. Thus, first-order QPTs can have a much higher sensitivity and provide a natural platform for quantum metrology,^1,11 quantum amplification,¹⁶ and new types of single-photon detectors.⁶ We note that, originally, the critical point has been specifically used to denote the point where a continuous thermodynamic phase transition occurs. Thus, the concept of the quantum critical point and quantum criticality is traditionally used only for continuous quantum phase transitions.⁴⁵ To avoid confusions in the interpretation of “critical” for discontinuous QPTs in the detector context vs the concept of “critical” in the continuous QPT context, we denote the optimum bias point of the detector as the phase transition point hereafter. It should, however, be noted that, even at the discontinuous first-order QPT point, the free energy and entropy still change continuously with temperature.

We also emphasize that the discontinuous jump and the diverging sensitivity in first-order QPTs result from a macroscopic order change in the ground states of two neighboring quantum phases. Due to the vanishing energy gap at the phase transition point, the transition between the ground states of two quantum phases cannot be realized by a unitary adiabatic operation.⁶¹ By varying a parameter across the phase transition point with time, one cannot induce a QPT from one ground state to the other in practice. Thus, a dynamical detection event may have totally a different sensitivity scaling. A few first-order QPTs have already been found, like the Dicke-Ising model,^31,62,63 the antiferromagnetic Ising chain,^64–66 the LMG model in the zero-field limit,⁶⁷ the quantum Ising model with a four-spin exchange interaction,⁶⁸ etc. Nevertheless, the dynamics of these QPTs around the phase transition point has not been revealed. The application of these first-order QPTs is, therefore, an open problem, as practical detection events and amplifications are fundamentally dynamical processes.

In the following, we will present our prototype QCD in detail. First, we introduce the model Hamiltonian of our QCD. Via the mean-field theory, we derive the complete phase diagram of the detector. We also show the details of the QPTs within the detector with numerical simulations, specifically the fundamental change in the ground-state wave function. Finally, we display the dynamical change in the OPs and the wave function of the detector to show the quantum critical amplification process within the detector.

In our previous work,⁹ we introduced a first-order QPT model, which is the underlying reason for amplification in the QCD. This model is composed of a bosonic mode and a spin ensemble with a homogeneous long-range dipole-dipole interaction along only one direction (⁠$y$-direction). Here, we extend this model to a more general case with three-dimensional homogeneous couplings (see Fig. 2),

Here, $d^(d^†)$ denotes the output bosonic mode. Its frequency has been taken as the unit of the energy $ω0=1$⁠, and all the other parameters in the Hamiltonian have been rescaled by $ω0$⁠. The operators $σ^jα(α=x,y,z)$ are the Pauli matrices of the $j$th spin. A magnetic field is applied along the $z$-direction inducing an energy splitting $ϵ$ between spin states $|↑⟩j$ and $|↓⟩j$⁠. The coupling between the bosonic mode and the spin ensemble is along the $x$-direction with homogeneous coupling strength $λ$⁠. The last term characterizes the all-to-all homogeneous dipolar coupling $J~α$ between the spins, which is significantly different from the nearest-neighbor coupling in the traditional Ising models.⁴⁵ 

After defining the collective angular momentum operators of the $N$ spins

we re-express our model Hamiltonian as

Here, we have used the angular momentum conservation relation $S^x2+S^y2+S^z2=N(2N+1)/4$ and defined the new homogeneous dipolar coupling strength $Jx=J~x−Jz~$ and $Jy=J~y−Jz~$⁠. Since the total angular momentum of the spins is conserved, we can perform the calculation in a subspace spanned by the Dicke states.⁶⁹ Then, the $N$-spin ensemble is now equivalent to a single particle with spin-$N/2$⁠. For simplicity, we only consider the case of ferromagnetic coupling and all the parameters in the Hamiltonian (3) are positive real numbers. The rich physics of the antiferromagnetic coupling model (with negative dipolar coupling $Jx$ and/or $Jy$⁠) will not be present in this paper. The first three terms compose the traditional Dicke model without the rotating-wave approximation,⁶⁹ and the last three terms form the well-known LMG model.^28–30 Hence, we call this model the Dicke-LMG model.

Our detector model (3) is not a simple combination of the Dicke model and the LMG model. New quantum phases and new QPTs, especially a first-order QPT, emerge in this model. For second-order QPTs, substantial changes occur only in the ground-state wave function, which makes it extremely hard to detect the criticality.^70,71 Specifically, single-shot deterministic readout is required for pulse signal detection instead of multiple repetitive probabilistic measurements. However, for first-order QPTs, discontinuous changes exist in directly measurable quantities at the phase transition point, which makes the quantum singularity significantly more detectable. Thus, first-order QPTs in a many-body system are of fundamental interest and possess much more application potential in quantum metrology and quantum detection.

In the traditional Dicke model,⁶⁹ a second-order superradiant QPT occurs when the spin-boson coupling exceeds the QPT point $λc,II≡ϵ/2$⁠.^{27,58–60,72} However, this superradiant QPT will be forbidden by the Thomas-Reiche-Kuhn (TRK) sum rule in a natural-atom system^33,34 or an artificial qubit system.⁷³ However, this problem is easily overcome in our model with the help of the dipolar interaction $Jx$ along the $x$-axis.

In the LMG model, second-order magnetic QPTs between the paramagnetic phase and the ferromagnetic phase have been found.^28–30 In 2004, Vidal et al. show the existence of a first-order QPT in the LMG model in the zero-magnetic-field limit $ϵ→0$⁠.⁶⁷ However, in this paper, we will show that the first-order QPT exists even for the finite magnetic field (⁠$ϵ≠0$⁠) case. This first-order QPT results from the competition between two ferromagnetic phases and can be exploited for quantum amplification. In the previous literature, this first-order QPT has not been revealed because the traditional magnetic OP (the mean magnetization of the spins) is selected. In the following, we will show that this magnetic OP cannot characterize the first-order QPT and we will introduce two new magnetic OPs to characterize the quantum phases in our model and to demonstrate the corresponding QPTs.

The key element of our QCD is the first-order-QPT-based quantum amplification. To utilize a QPT system as a novel amplification source, one first needs a clear phase diagram and the explicit boundaries between the quantum phases of a system. In this section, we introduce two magnetic OPs to characterize the first-order QPT of the interacting spin system. Then, we present the complete phase diagram of the Dicke-LMG model (3), which is obtained by employing a mean-field theory.

In different phases, matter usually has different long-range orders. Thus, an OP is utilized to measure the degree of order in a phase transition system, such as the density change in solid-liquid-gas transitions. The OPs normally range between zero in one phase and nonzero in the other.

To characterize the first-order magnetic QPTs, we now introduce two new magnetic OPs,

and

In the ground state of the paramagnetic phase, all the spins are polarized along the negative $z$-axis, i.e., $|↓↓↓⋯↓⟩z$⁠. In this state, the mean values of these magnetic OPs are zero. When the system goes to the ferromagnetic phase, the magnetic OPs increase from zero to a finite value due to the macroscopic polarization of spins in the $xy$-plane.

These two magnetic OPs actually characterize the spin fluctuations in the $xy$-plane,

and

Here, we have used the fact that the mean magnetizations in the $xy$ plane $Mx≡⟨S^x⟩0/N$ and $My≡⟨S^y⟩0/N$ are zero in the ferromagnetic phases. Because the two degenerate ground states $|→→→⋯⟩$ and $|←←←⋯⟩$ have opposite polarizations.^45,74,75 The change in the transverse magnetic fluctuations during a magnetic QPT can be probed experimentally through spin noise spectroscopy.⁵³ 

The superradiant phase can be well characterized by the superradiant OP, which is defined as the rescaled excitation number in the bosonic mode,^27,58

According to the Ginzburg–Landau theory, we should choose the spontaneous magnetization in the $xy$-plane (i.e., $ζM,x$ and $ζM,y$⁠) as the OPs for magnetic phase transition^76,77 and the amplitude of the bosonic mode (⁠$ζS$⁠) as the OP for the superradiant phase transition. However, we do not use these traditional OPs for the following three reasons: (1) we find that the high-order correlation $⟨S^x2S^y2⟩0$ diverges at the first-order magnetic phase transition point, but $⟨S^xS^y⟩0$ shows no singularity;⁴⁹ (2) $ζM,x$ and $ζM,y$ are equivalent to $⟨σ^ixσ^i+1x⟩0$⁠, which is usually selected as the OP of the Ising spin chain;⁴⁵ (3) usually, the particle number of the bosonic mode is measured in the experiment instead of its amplitude.

Next, we calculate the mean values of these OPs in each quantum phase via a mean-field theory and then we show the full phase diagram of the Dicke-LMG model.

A many-body system with interactions is generally very difficult to solve exactly. The mean-field theory was invented to address this problem. In this method, the Hamiltonian is expanded in terms of the magnitude of fluctuations around the mean of the operators. The mean-field theory is the zeroth-order expansion of the Hamiltonian. The minimum of the mean-field value of the Hamiltonian gives the ground-state energy of the system. The first-order term will be of scale $1/N$⁠, which can be neglected in the large-$N$ limit for the ground-state problem. Recently, Zhang et al. presented an elegant mean-field theory to calculate the quantum phases in a many-body system.⁶⁶ In this subsection, this method is exploited for the Dicke-LMG model in (3) to obtain the ground state of the whole system and the phase boundary between different quantum phases. We will also verify this mean-field phase diagram with exact numerical simulation in Sec. V.

The spin-boson coupling in our Hamiltonian (3) functions as a displacement of the bosonic mode. Therefore, in a mean-field ground state, the operator $d^$ can be replaced with a complex number $Nα$ by assuming that the bosonic mode is a coherent state $|Nα⟩$⁠.⁷⁸ The ground state of the LMG model in the thermodynamic limit $N→∞$ is proven to be a coherent spin state.^79,80 Thus, it is reasonable to replace the spin operators $(σ^jx,σ^jy,σ^jz)$ with a classical Bloch vector, $n→=(sin⁡θcos⁡ϕ,sin⁡θsin⁡ϕ,cos⁡θ)$⁠.^81,82

To be consistent with the mean-field theory, we modified the definition of the coherent spin state^83,84 as

Thus, the angle $θ$ is identical to the polar angle of a spherical coordinate. Here, $|N/2,N/2⟩$ is the Dicke state with all spins in the up state $|↑⟩z$⁠. In the following, we denote the ground state of the whole system as $|Nα⟩⊗|θ,ϕ⟩,$ i.e., the direct product of a bosonic coherent state $|Nα⟩$ and a coherent spin state $|θ,ϕ⟩$⁠.

The scaled ground-state energy $E(α,θ,ϕ)=⟨H⟩0/N$ is given by

where a constant has been neglected. To minimize the ground-state energy, we need to set the first derivatives of $E$ with respect to $α$⁠, $α∗$⁠, $θ$⁠, and $ϕ$ to zero. Since all the parameters in Hamiltonian (3) are assumed to be positive, it is easy to verify that, in the ground state, the amplitude of the bosonic mode should be real, i.e., $Imα=0$⁠. Then, minimization conditions give the following equilibrium constraints:

where $θ∈[0,π]$ and $ϕ∈[0,2π)$⁠. To guarantee that the minimums of $E(α,θ,ϕ)$ are obtained, one needs to calculate the second derivatives. The ground-state stability is determined by a $3×3$ Hessian matrix

The ground states are stable only if $M$ is positive definite, i.e., all eigenvalues of $M$ are non-negative.

We now construct the phase diagram with the help of the three OPs we defined

The magnetization in the $z$-direction can also be calculated easily

According to the values of the OPs, we find three quantum phases in the Dicke-LMG model: paramagnetic-normal (PN) phase, ferromagnetic-normal (FN) phase, and ferromagnetic-superradiant (FS) phase. In Table I, we list the stability conditions (phase boundaries), ground states, and the mean value of the three OPs on the ground state of each phase. More details can be found in  Appendix A. We emphasize that the first-order magnetic QPT can be properly characterized only by the two magnetic OPs we introduced.

To clearly reveal the relationship between these quantum phases, we display the phase diagram of the Dicke-LMG model in Fig. 3. In panel (a), we plot the phase diagram in the $(2λ2+Jx,Jy)$ plane and also show the ground state of each phase. The boundaries between the three quantum phases are depicted by the blue, green, and red lines. There exists a unique triple-point $(2λ2+Jx=ϵ/2,Jy=ϵ/2)$⁠, where the three lines intersect. A second-order QPT from the PN phase to the FN phase occurs when increasing the spin-spin coupling $Jy$ across the blue line $Jy=ϵ/2$⁠. The magnetic OP $ζM,y$ changes from $0$ to a finite value continuously after the QPT. A second-order QPT from the PN phase to the FS phase happens when increasing the spin-boson coupling $λ$ (or spin-spin coupling $Jx$⁠) across the green line $2λ2+Jx=ϵ/2$⁠. Both the superradiant OPs (⁠$ζS$⁠) and the magnetic OP (⁠$ζM,x$⁠) change from $0$ to a finite value continuously during this QPT. Particularly, the QPT from the FN phase to the FS phase, crossing the red line $2λ2+Jx=Jy$⁠, is of a first-order transition. This occurs in the strong-coupling region when both the $x$-direction coupling $2λ2+Jx>ϵ/2$ and the $y$-direction coupling $Jy>ϵ/2$ exceed the second-order QPT points. The OP $ζM,y$ suddenly drops from a finite value to zero and, at the same time, both $ζS$ and $ζM,x$ discontinuously jump from zero to finite values. This first-order QPT plays a crucial role in quantum amplification of our QCD.

We emphasize that the first-order QPT in our detector model fundamentally results from the competition between two ferromagnetic phases. To clarify this, we draw the same phase diagram in the $(2λ2+Jx,ϵ)$-plane in panel (b). As the $ϵ$-axis is parallel to the first-order QPT boundary (the red line), the first-order QPT will not happen if we only vary the magnetic field strength $ϵ$⁠. We can also verify that the first-order phase transition represented in the previous Dicke-Ising model⁶² has the same origin as the one we addressed here. We predict that there is a universal first-order QPT in interacting spin systems from the nearest-neighbor short-range coupling to the all-to-all long-range coupling,

where $⟨i<j⟩$ run for the $n$ nearest neighbors and $n=1$ for the Ising XY model and $n=(N−1)≈N$ for the LMG model. The first-order QPT in this interacting spins model can be verified with the mean-field theory addressed above.

In Sec. V, we numerically demonstrate all the QPTs existing in our detector model and display the fundamental changes in the ground-state wave function during the QPTs with Husimi $Q$-functions.

To reveal the microscopic mechanism of amplification in the first-order-QPT-based QCD, we first need to understand the underlying physics of the QPTs in the model, especially the fundamental change in the ground state wave function of the system during the QPTs. In the Dicke-LMG model, QPTs occur at the boundaries of the phase diagram given in Fig. 3. To deeply understand all the quantum phases and classify the order of the QPTs correctly, we split the full model in Eq. (3) into three submodels: the LMG model, the Dicke-LMGx model, and the Dicke-LMGy model. In Secs. V A–V C, we will study these submodels separately. We follow a simple recipe, first present the numerical demonstration of the phase diagram, then show the details of the QPTs, and finally display the quasiprobability distribution of the ground states.

In this subsection, we first consider the simplest submodel—the LMG model with vanishing spin-boson coupling (⁠$λ=0$⁠) in Hamiltonian (3),

The LMG model was first proposed to describe the shape phase transition in nuclei.^28–30 As an exactly solvable QPT model,^85–88 the LMG model has attracted significant attention in ground-state entanglement,^67,89 spin squeezing,^90,91 criticality detection,^92,93 heat-engine efficiency enhancement,⁹⁴ dynamics of the QPT,⁹⁵ etc. However, the first-order QPT in the LMG model has not been fully revealed due to the improperly selected magnetic OP as the net magnetization $Mz$⁠.^67,89 Here, with the help of the magnetic OPs $ζM,x$ and $ζM,y$⁠, we introduced in Sec. IV, we numerically show the first-order QPT as well as the well-understood second-order ones. Utilizing the Husimi spin $Q$-function, we also verify that the ground state of the LMG model is indeed a coherent spin state.

We now show the complete phase diagram of the LMG model, which has not been revealed in the previous literature. The phase diagram of the LMG model extracted from the mean-field theory is displayed in Fig. 4(a), which can also be obtained by setting $λ=0$ in Fig. 3. The LMG model has one paramagnetic (PM) phase and two ferromagnetic phases (FM-X and FM-Y). The numerical demonstrations of the phase diagrams are shown in Figs. 4(b) and 4(c). If both the spin-spin coupling in the $x$-direction (⁠$Jx<Jxc,II$⁠) and the $y$-direction (⁠$Jy<Jyc,II$⁠) are below their corresponding phase transition coupling strengths (⁠$Jxc,II≡ϵ/2$ and $Jyc,II≡ϵ/2$⁠), both the two magnetic OPs (⁠$ζM,x$ and $ζM,y$⁠) are zero, and thus the spins are in the PM phase. The well-known second-order QPT from the PM phase to the FM-X (FM-Y) phase occurs when $Jx$ (⁠$Jy$⁠) crosses the phase boundary denoted by the green (blue) line in panel (a). However, in the strong spin-spin coupling regime with $Jx>Jxc,II$ and $Jy>Jyc,II$⁠, the sharp changes in the OPs at the boundary $Jx=Jy$ indicate a first-order QPT, which has not been revealed in the previous literature. This first-order QPT can also be verified in experiments by measuring the change in the magnetic fluctuations along $x$ and $y$ directions. We also plot the net magnetization in the $x$-direction in panel (d). With the two magnetic OPs we introduced, we can easily verify that the anisotropic transition in the Ising $XY$ chain⁹⁶ is also a first-order QPT.

To reveal the details of the QPTs, we plot the magnetic OPs as functions of spin-spin coupling $Jx$ for different spin-spin coupling $Jy$ in Fig. 5. From the red-solid and green-dashed lines in Fig. 5(a), we can see that second-order QPTs occur at $Jxc,II≡ϵ/2$ when the spin-spin coupling along the $y$-axis is below the phase transition point $Jy≤Jyc,II≡ϵ/2$⁠. If we fix $Jx≤Jxc,II≡ϵ/2$ but increase $Jy$ across the phase transition point $Jyc,II$⁠, similar second-order QPTs (data not shown) from the PM phase to the FM-Y phase also occur. In the strong spin-spin coupling case when $Jy$ is larger than the second-order QPT strength $Jy>Jyc,II$⁠, a QPT occurs at a new phase transition point $Jxc,I≡Jy$ [see the blue-dotted line in Fig. 5(a)], which we verify as the first-order later. In Fig. 5(b), we show that, only after a first-order QPT occurs, there exists a sudden drop in the magnetic OP $ζM,y$ at the phase transition point $Jxc,I$ (blue-dotted line), as the system transitions from the FM-Y phase to the FM-X phase. Thus, this first-order QPT results from the competition between the two ferromagnetic phases. In Fig. 5(c), we show that the first-order QPT cannot be revealed by the traditional magnetic OP $Mz$⁠, which changes continuously in both second- and first-order QPTs.

To verify the first-order magnetic QPT, we plot the magnetic OP $ζM,x$ for different spin numbers $N$ but with fixed $Jy=0.6>Jyc,II$ in Fig. 5(d). The slope of $ζM,x$ at the phase transition point $Jxc,I=Jy$ increases with spin number $N$⁠. Now, we define the sensitivity function of the LMG model as the first derivative of the magnetic OP $ζM,x$⁠, i.e., the sensitivity,

From Fig. 6(a), we see a very sharp peak located at the first-order QPT point (the blue-dashed line), which is obtained with strong spin-spin coupling $Jy>Jyc,II$⁠. The peaks of the sensitivity function $χ(Jx)$ for second-order QPTs (red and blue curves) are much lower than the first-order QPT. The position of the maximum of $χ$ for second-order QPTs will approach the phase transition point $Jxc,II$ asymptotically in the thermodynamic limit $N→∞$⁠, but the height converges to a finite value. In Fig. 6(b), we plot the maximum sensitivity $χmax$ of the first-order QPT for different spin numbers. We see that $χmax$ diverges with increasing spin number in $N2$ scaling. Similar $N2$ scaling will also be found later for the first-order QPT in the Dicke-LMGy model.

In the last part of this subsection, we shed light on the ground states of the three quantum phases. In 2005, Dusuel and Vidal studied the ground states of the LMG model in detail.⁸² Via the Holstein-Primakoff transform, they found that, in the thermodynamic limit $N→∞$⁠, the ground state of the LMG model is exactly a coherent spin state. Here, we numerically verify this result. Particularly, we show the fundamental changes of the ground-state wave functions during magnetic QPTs via the spin Husimi $Q$-function,⁹⁷ 

where $ρ^g$ is the ground-state density matrix of the spins. To vividly show the quasiprobability distribution of the ground state wavefunction, we transfer the spherical coordinates $(r,θ,ϕ)$ to the corresponding Cartesian coordinates $(x,y,z)$⁠. Here, $r=Q(θ,ϕ)/max[Q(θ,ϕ)]$ is the normalized $Q$-function, which is extensively used to characterize the spin squeezing effect.⁹⁸ 

The ground state of the PM phase is the Dicke state $|N/2,−N/2⟩$ (i.e., the coherent spin state $|θ0,ϕ0⟩$ with $θ0=π$ and totally undetermined $ϕ0$⁠). In this state, all the spins are polarized along the negative $z$-axis. As shown in Fig. 7(a), the corresponding $Q$-function is a cigarlike structure lying along the negative $z$-axis. The cross sections of the $Q$-function in the $xz$ and $yz$ planes are displayed in Figs. 7(b) and 7(c), respectively. The red lines originate from the numerical ground states obtained by directly diagonalizing the LMG model and the gray lines are the analytic results of the Dicke state $|N/2,−N/2⟩$⁠. The numerical and analytical results exactly coincide with each other. From this quasiprobability distribution function, we can also see that the mean magnetization $Mz$ is a finite negative value, but the two order parameters $ζM,x$ and $ζM,y$ are negligibly small and approach zero in the limit $N→∞$⁠.

For the FM-X phase, there exist two degenerate ground states $|θ0,0⟩$ and $|θ0,π⟩$⁠,⁸² where the polar angle $θ0$ is determined by the parameters of the Hamiltonian as given in Eq. (A14). The system can be in an arbitrary superposition of these two degenerate states. Thus, the ensemble mean of the magnetization along the $x$-direction $⟨S^x⟩0$ is zero, but the magnetic fluctuations characterized by the magnetic OP $ζM,x$ is finite. The numerical simulation of the Husimi $Q$-functions of states $|θ0,π⟩$⁠, $|θ0,0⟩$⁠, and their symmetric quantum superposition $|G+⟩=(|θ0,0⟩+|θ0,π⟩)/2$ are presented by the three columns in Fig. 8, respectively. The cigarlike structures lie in the $xz$-plane, as the strong spin-spin coupling $Jx$ along the $x$-direction overwhelms the spin-spin coupling along the $y$-direction in the FM-X phase. From the $Q$-function, we can also see that $Mz$ is a finite negative value. The magnetic OP $ζM,y$ is very small and will approach zero when $N→∞$⁠. In the bottom row, we show the cross section of the $Q$-functions in the $xz$-plane. The blue-solid lines are obtained by the numerical ground states, and the gray-dotted lines are the analytic results for the corresponding three coherent spin states. The perfect coincidence verifies that the ground state of the LMG model in the FM-X phase is indeed a coherent spin state in the large $N$ limit.

Similar to the FM-X phase, the FM-Y phase also has two degenerate ground states $|θ0,π/2⟩$ and $|θ0,3π/2⟩$⁠, where the polar angle $θ0$ is determined by the parameters of the Hamiltonian as given in Eq. (A10). The ensemble mean value of both $⟨S^x⟩0$ and $⟨S^y⟩0$ are zero. However, the magnetic fluctuations along the $y$-axis characterized by the magnetic OP $ζM,y$ is finite, which can be seen from the $Q$-function of states $|θ0,π/2⟩$⁠, $|θ0,3π/2⟩$⁠, and $|G+⟩=(|θ0,π/2⟩+|θ0,3π/2⟩)/2$ in Fig. 9. The cigarlike structures lie in the $yz$-plane, as the strong spin-spin coupling $Jy$ along the $y$-direction dominates in the FM-Y phase. In this case, $ζM,x$ is very small and goes to zero for $N→∞$⁠.

In summary, we have revealed the first-order QPT in the traditional LMG model via the magnetic OPs $ζM,x$ and $ζM,y$⁠. We predict that a similar first-order QPT should also exist in the XY Ising chain in a transverse field. The Husimi $Q$-functions show the fundamental difference between the ground state wave functions of the three quantum phases in the LMG model. We also numerically verified that the ground states of the LMG model are coherent spin states.

In this subsection, we mainly focus on the second-order superradiant QPT in the model,

This model can be obtained by setting the spin-spin coupling along the $y$-direction to zero (i.e., $Jy=0$⁠) in the Dicke-LMG model (3). Thus, we call it the Dicke-LMGx model. In the original Dicke model,⁶⁹ Dicke investigated the superradiance of an atomic ensemble composed of a large amount of indistinguishable two-level atoms as opposed to spins. Nevertheless, mathematically, indistinguishable two-level atoms are equivalent to spin-halves. Here, for the sake of consistency, we replace the atoms with spins.

The superradiant QPT in the Dicke model was first predicted by Hepp and Lieb in 1973.²⁷ Via analyzing the free energy per particle, they found that there exists a thermodynamic phase transition as well as a QPT from the normal phase to the superradiant phase, in which macroscopic excitations exist in the ground state. Later, Wang and Hioe theoretically revisited this issue with a much simpler method.⁵⁸ The QPT is easily shown with numerical simulation and has also been demonstrated in a recent experiment with a Bose-Einstein condensate (BEC).⁹⁹ However, the thermodynamic phase transition has not been demonstrated and the underlying mechanism is also missing. During our numerical simulation, we found that the phase transition temperature is strongly dependent on the order of two limits: the thermodynamic limit $N→∞$ and the Hilbert space cutoff corresponding to the bosonic mode $Ncutoff→∞$⁠. In the following, we only focus on the superradiant QPT.

There is a fundamental challenge to achieve the superradiant QPT in the Dicke model—the no-go theorem. The occurrence of the superradiant QPT is forbidden by the Thomas-Reiche-Kuhn (TRK) sum rule,^33,34 as the atom-field (the spin-boson in our model) coupling cannot exceed the phase transition strength $λc,II≡ϵ/2$ in a natural atomic system. There are many theoretical proposals to circumvent the no-go theorem. In 2007, Dimer et al. proposed to construct an effective Dicke Hamiltonian with a four-level atomic ensemble in an optical cavity.¹⁰⁰ Ciuti et al. proposed Cooper pair boxes capacitively coupled to a resonator⁷³ or three-level systems¹⁰¹ to realize the superradiant QPT. Bastidas et al. suggested periodically modulating the spin-boson coupling to circumvent the no-go theorem.¹⁰² Recently, Lü et al.¹⁰³ utilized the nonlinear coupling between a mechanical oscillator and an ancillary cavity mode to realize the superradiant QPT in the mechanical mode. In this subsection, we will show that this challenge can be easily overcome with the help of the dipole-dipole interaction between spins (atoms), which has previously been disregarded. The spin-spin coupling lowers the phase transition spin-boson coupling strength required by the superradiant QPT.

The Dicke-LMGx model has two quantum phases: the PN phase and the FS phase. The phase diagram of the Dicke-LMGx model can be obtained by setting $Jy=0$ in Fig. 3. We present the numerical simulation of the phase diagram with the superradiant OP $ζS$ and the magnetic OP $ζM,x$ in panels (a) and (b) in Fig. 10, respectively. The other magnetic OP $ζM,y$ is zero in both phases and has been ignored in this subsection. The phase boundary given by $4λ2+2Jx=ϵ$ is depicted by the green line, which corresponds to the green line in Fig. 3 exactly. In the PN phase, both the superradiant OP $ζS$ and the magnetic OP $ζM,x$ are zero. In the FS phase, both $ζS$ and $ζM,x$ are finite except when $λ=0$⁠. In panel (a), we see close to the $Jx$ axis, $ζS$ marginally increases with $Jx$ after passing the boundary and its magnitude is negligibly small even in the FS phase, as the magnetic OP $ζS$ is proportional to $λ2$ [see Eq. (A17)]. However, $ζM,x$ always increases to a relatively large value after the parameters cross the phase boundary as shown in panel (b).

Now, we show how to overcome the no-go theorem in superradiant QPT. In Fig. 11, we plot the OPs $ζS$ and $ζM,x$ in panels (a) and (b) as functions of spin-boson coupling $λ$ with fixed spin-spin coupling $Jx$⁠. The position of the new phase transition spin-boson coupling $λc,II≡ϵ−2Jx/2$ is marked by the vertical black-dashed lines. For a larger $Jx$ case, a relatively smaller spin-boson coupling $λ$ is required to trigger the QPT from the PN phase to the FS phase. With the help of the dipolar coupling along the $x$-axis, a weak spin-boson coupling can still induce the superradiant QPT. Thus, the constraint from the no-go theorem in the Dicke model is easily overcome. However, to obtain a larger superradiant OP $ζS$ (more excitations in the bosonic mode), strong spin-boson coupling is still needed. In Fig. 12, we plot the OPs $ζS$ and $ζM,x$ as functions of spin-spin coupling $Jx$ with fixed spin-boson coupling $λ$ in panels (a) and (b), respectively. After the QPT, smaller $ζS$ are obtained for smaller $λ$ but with the same $Jx$⁠. Specifically, in the case of $λ=0$⁠, the superradiant OP $ζS$ is still zero after the QPT when $Jx>Jxc,II$ as shown by the red line in panel (a). In this case, the system is actually in the FN phase.

At the end of this subsection, let us shed light on the ground states of the system with the Husimi $Q$-functions. The $Q$-function of the spin is defined in Eq. (22) after tracing out the bosonic degrees of freedom, such that the $Q$-function of the bosonic mode is defined as

where $ρ^g$ is the ground-state density matrix of the entire system, $|α⟩$ is the bosonic coherent state with amplitude $α$⁠, and $Trspin$ means tracing out the spin degrees of freedom. The ground state of the PN phase is $|0⟩⊗|π,ϕ0⟩$⁠, i.e., the bosonic mode is in the vacuum state and the spins are in the coherent spin state with $θ0=π$ and undetermined $ϕ0$⁠. The FS phase has two degenerate ground states $|Nα0⟩⊗|θ0,π⟩$ and $|−Nα0⟩⊗|θ0,0⟩$⁠, where $θ0$ and $α0$ are determined by the parameters of the system as given in Eqs. (A14)–(A16). In the FS phase, the system can be in an arbitrary superposition of these two degenerate states. Thus, the ensemble mean values of $⟨d^†⟩0$ (or $⟨d^†⟩0$⁠) and $⟨S^x⟩0$ are zero. However, the mean of the coupling terms $⟨d^S^x⟩0$ and $⟨d^†S^x⟩0$ are finite negative values, which minimize the ground-state energy of the whole system.

The ground state of the bosonic mode changes from the vacuum state to a coherent state after the superradiant QPT. The $Q$-function of the bosonic mode for different spin-boson couplings is displayed in Fig. 13. In panel (a), we show the $Q$-function of the bosonic mode in the ground state (the vacuum state) of the PN phase with $2Jx+4λ2<ϵ$⁠. There is only one peak located at $α=0$⁠. For panels (b) and (c), the system is in the FS phase with $2Jx+4λ2>ϵ$ and the spin-boson coupling $λ$ leads to shifting of the peak along the real axis (as proven in Sec. IV). Here, we take the ground state of the FS phase as the symmetric quantum superposition state $|G+⟩=(|Nα0⟩⊗|θ0,π⟩+|−Nα0⟩⊗|θ0,0⟩)/2$⁠. In panel (b), the spin coupling $λ$ is not large enough to split the $Q$-function into two separated peaks like in panel (c). This demonstrated more in panel (d), which shows the cross section of the $Q$-function on the real-$α$ plane.

The $Q$-function of the spins is displayed in Fig. 14 with the same parameters as in Fig. 13. For the PN phase as shown in panel (a), the normalized spin $Q$-function is a cigarlike structure lying along the negative $z$-axis. For the FS phase as shown in panels (b) and (c), the interaction along the $x$-axis splits the $Q$-function in the $xz$-plane. The $Q$-function of the ground state $|G+⟩$ has two branches on the negative-$x$ and positive-$x$ half planes corresponding to the two degenerate ground states, respectively. For larger $λ$⁠, larger polar angles $θ0$ are obtained as shown in panel (d).