Quantum Fisher information as signature of superradiant quantum phase transition

The single-mode Dicke model is well-known to undergo a quantum phase transition from the so-called normal phase to the supperradiant phase (hereinafter called the"superradiant quantum phase transition"). Normally, quantum phase transitions are closely related to the critical behavior of quantities such as entanglement, quantum fluctuations, and fidelity. In this paper, we study quantum Fisher information (QFI) of the field mode and that of the atoms in the ground state of the Dicke Hamiltonian. For finite and large enough number of atoms, our numerical results show that near the critical atom-field coupling, the QFIs of the atomic and the field subsystems can surpass the classical limits, due to the appearance of nonclassical squeezed states. As the coupling increases far beyond the critical point, the two subsystems are in highly mixed states, which degrade the QFI and hence the ultimate phase sensitivity. In the thermodynamic limit, we present analytical results of the QFIs and their relationships with the reduced variances. For each subsystem, we find that there is a singularity in the derivative of the QFI at the critical point, a clear signature of quantum criticality in the Dicke model.


I. INTRODUCTION
Quantum phase transitions in many-body systems are of fundamental interest [1] and have potential applications in quantum information [2][3][4][5][6][7] and quantum metrology [8][9][10][11][12][13][14][15]. Consider, for instance, a collection of N two-level atoms interacting with a single-mode bosonic field, described by the Dicke model (with = 1 throughout this paper) [16]: whereb andb † are annihilation and creation operators of the bosonic field with oscillation frequency ω, which is nearly resonant with the atomic energy splitting ω 0 . The collective spin operatorsĴ ± ≡Ĵ x ± iĴ y = kσ ± k andĴ z = kσ z k /2 obey the SU(2) Lie algebra, whereσ ± k andσ z k are Pauli operators of the k-th atom. The atom-field coupling strength λ ∝ √ N/V depends on the atomic density N/V. For a finite number of atoms N (≡ 2 j), the Hamiltonian (1) commutes with the parity operatorΠ = exp[iπ(b †b +Ĵ z + j)], due toΠ †Ĵ xΠ = −Ĵ x and Π †bΠ = −b [17]. As a result, the ground state of the finite-N Dicke model |g does not exhibit any singularity and degeneracy. This can be understood by expanding |g in the basis {|n | j, m } [18,19], where |n and | j, m (with m ∈ [− j, + j]) are the Fock states and the eigenvectors ofĴ z , respectively. For vanishing atom-field coupling strength λ, the ground state |g = |0 | j, − j has a positive parity Π = +1; Similarly for λ > 0, due to the conserved parity, the ground state |g consists of states with even number n + m + j [18,19], which results in vanishing coherence (i.e., Ĵ x = b = 0). However, in the thermodynamic limit (for finite N/V as N, V → ∞), the parity symmetry is spontaneously broken and the ground states with parities ±1 become degenerate in the superradiant phase (i.e., the symmetry-broken phase at λ ≥ λ cr = √ ω 0 ω/2) [18][19][20][21][22][23][24][25][26][27][28][29][30], leading to bifurcation of Ĵ x and that of b [17]. Unlike the traditional phase transition of the Dicke model at a finite temperature [31], the superradiant quantum phase transition is driven by quantum fluctuations in the large-N limit. It is natural to ask in what different ways one can characterize such a quantum phase transition in a realistic system. Several quantities, with various degrees of experimental accessibility, have been shown to be sensitive to the quantum phase transitions, such as the von Neumann entropy [7], the fidelity [32], and more recently the quantum fluctuations of the field [33].
In this paper, we investigate the quantum Fisher information (QFI) of the field stateρ B = Tr A (|g g|) and that of the atomic stateρ A = Tr B (|g g|), where Tr A (Tr B ) is the partial trace of the ground state |g over the atomic (bosonic field) degrees of freedom. In quantum metrology, the QFI is one of central quantities to qualify the input state [34,35], especially in March-Zehnder (or, equivalently, Ramsey) interferometer-based phase or parameter estimation. The achievable phase sensitivity is well-known to be limited by the quantum Cramér-Rao bound δϕ min ∝ 1/ √ F, where the QFI F depends on the input state and the phase-shift generator [35][36][37]. Here, we show that near the critical point λ cr , the QFI ofρ A,B for the finite-N Dicke model can surpass the classical limit due to the nonclassical squeezed properties of the ground state. As the coupling strength λ ≫ λ cr , botĥ ρ A andρ B become highly mixed states, which leads to the QFI of the field returning to the classical limit, while for the atoms the QFI tends to be zero. In the thermodynamic limit, we discover that there exists analytical relationships between the QFIs and the reduced variances, which show clearly the squeezing-induced enhancement of the QFIs. More interestingly, we find that the derivative of F for each subsystem is di-vergent at λ = λ cr , similar to the fidelity of the ground state |g in the one dimensional Ising chain [32]. This finding suggests that the QFI could be useful as a sensitive probe of quantum phase transitions [12].

II. QUANTUM FISHER INFORMATION IN FINITE-N DICKE MODEL
We first examine the field stateρ B = Tr A (|g g|) of the finite-N Dicke model by numerically evaluating the QFI with respect toρ B (ϕ) = e iϕĜρ B e −iϕĜ , where ϕ is an unknown phase shift andĜ is the phase-shift generator (=b †b for the singlemode field [37]). In general, the field stateρ B (ϕ) is a mixed state and the QFI is given by [35][36][37][38][39][40] where the weights {p n } are nonzero eigenvalues ofρ B , and {|ψ n } are the corresponding eigenvectors. The first term of Eq.
(2) is a weighted average over the QFI for each pure state |ψ n , and the variance (∆Ĝ) 2 n ≡ ψ n |Ĝ 2 |ψ n − | ψ n |Ĝ|ψ n | 2 . The second term is simply a negative correction (c.f. Ref. [40]). For a pure coherent state |α , with mean number of bosons n = |α| 2 , we obtain the QFI of the bosonic field, denoted by F B hereafter, F B = 4(∆b †b ) 2 = 4n and hence the ultimate sensitivity δϕ cl min = 1/(2 √n ), which corresponds to the classical (or shot-noise) limit. A sub shot-noise-limited phase sensitivity with δϕ < δϕ cl min is achievable provided that F B > 4n, which has been shown a nonclassical criterion ofρ B for the single-mode linear interferometer [41].
The atoms in the ground stateρ A = Tr B (|g g|) can also be used as a probe in a standard Ramsey interferometer. Since the orientation of the atomic spin Ĵ is along theĴ z axis (due to Ĵ + = 0), to precisely estimate the atomic transition frequency ω 0 , a π/2 pulse is required to rotate the atomic spin about theĴ y axis. After a free evolution τ, the phase shift ϕ = ω 0 τ is accumulated, leading to the atomic statê ρ A (ϕ) = e iϕĴ z e iπĴ y /2ρ A e −iπĴ y /2 e −iϕĴ z , where e iπĴ y /2 and e iϕĴ z represent the action of the pulse and the phase accumulation, respectively. Again, the QFI of the reduced atomic stateρ A (ϕ) is given by Eq. (2), where the phase-shift generatorĜ is replaced byĴ x and {|ψ n } are eigenvectors ofρ A with nonzero weights p n . For a coherent spin state | j, − j = | ↓ ⊗N , we have the QFI of the atoms F A = 4(∆Ĵ x ) 2 = N so the sensitivity is limited by δϕ cl min = 1/ √ N (i.e., the classical limit). Hereafter, we denote F A as the QFI of the atoms, to distinguish it from that of the bosonic field F B .
In Fig. 1, we plot the scaled QFI of the field F B /(4n) and that of the atoms F A /N as a function of the atom-field coupling strength λ. The QFI of the field vanishes at the atomfield coupling λ = 0, since the bosonic field is in vacuum (i.e., ρ B = |0 0|). By contrast, the QFI of the atoms is given by F A = N for the coherent spin stateρ A = | j, − j j, − j|, as mentioned above. When the coupling λ increases up to its critical point λ cr , a large number of bosons appears [18,27] and F B begins to increase. It surpasses the classical limit around λ cr as the ratio F B /(4n) > 1 [see the solid lines of Fig. 1(a)]. From Fig. 1(b), one can note that the QFI ofρ A with small N cannot beat the classical limit; The ratio F A /N is always smaller than 1 and decreases monotonically with increasing λ. Only for large enough number of atoms (say, N > 10), the scaled QFI F A /N can be larger than 1 at λ ∼ λ cr .
FIG. 1: (color online) Scaled quantum Fisher information of the bosonic field F B /(4n) (a) and that of the atoms F A /N (b) as a function of the coupling strength λ for a finite number of atoms N = 2, 6, 10, and 20, as indicated by the arrow. Horizontal dotted lines: the classical (or shot-noise) limit for the field mode F B = 4n (with mean number of bosonsn) and that of the atoms F A = N. Dashed lines: analytical results of the QFIs in the thermodynamic limit (i.e., N = ∞). For each stateρ A,B , the derivative of the QFI has a singularity at the critical point λ cr . Other parameters: the critical coupling It is interesting to observe two key features of the finite-N Dicke model: (i) near the critical point λ cr , bothρ A andρ B provide enhanced QFIs beyond the classical limits, although they are in general highly mixed states; (ii) for λ ≫ λ cr , the QFI of the field approaches the classical limit, i.e., F B → 4n, while for the atoms, F A → 0. To understand these behaviors, we study in detail the quantum nature ofρ A,B (see below). In Sec. III, we further present analytical results of the QFIs in the thermodynamic limit (i.e., N = ∞), and find that both F B /(4n) and F A /N show critical behaviors at λ = λ cr .
To confirm the presence of nonclassical states at λ ∼ λ cr , we consider the quadrature squeezing of the field stateρ B , following the original calculations by Emary and Brandes [18]. As usual in quantum optics, we introduce a quadrature operator where the squeezing angle σ ∈ [0, π/2] is to be determined.
When σ = 0 or π/2, the quadrature operator represents the amplitude or the phase component of the field mode, i.e., For the vanishing coupling λ, the field is in the vacuum |0 and hence the variance (∆X σ ) 2 = 1/4, which is the classical limit of the field variance and is independent of the squeezing angle σ. This isotropic variance has been depicted in the right panel of Fig. 2(a). As the coupling λ increases, one finds where we have used b = 0 and b 2 ∈ R, due to the parity symmetryΠ †bΠ = −b and the real atom-field coupling λ. Minimizing (∆X σ ) 2 with respect to σ, we obtain the optimal squeezing angle σ op = 0 or π/2. Our numerical result in Fig. 2(b) suggests σ op = π/2, which means that the optimal squeezing occurs along theX π/2 axis with the reduced variance (∆X π/2 ) 2 smaller than the classical limit 1/4. In Fig. 3(a), we confirm that the degree of squeezing 4(∆X π/2 ) < 1 and that it is minimized at λ ∼ λ cr for large enough N.
FIG. 3: (color online) Degree of quadrature squeezing for the field mode 4(∆X π/2 ) 2 (a), and that of spin squeezing for the atoms ξ 2 (b) against the coupling strength λ for the number of atoms N = 2, 6, 10, and 20, as indicated by the arrow. Dashed lines: analytical results in the thermodynamic limit (i.e., N = ∞), minimized at the critical point λ cr = 0.5 (on resonance, as Fig. 1), which indicates that the field and the atomic statesρ B,A are nonclassical phase-squeezed states near the critical point.

III. QUANTUM FISHER INFORMATION IN THE THERMODYNAMIC LIMIT
In this section, we first briefly review the quantum critical behavior of the Dicke model in the thermodynamic limit based on the solution outlined by Emary and Brandes [18], and then present analytical results of the QFIs for the field and the atomic subsystems.
The two possible shifts in Eq. (4) correspond to opposite spatial displacements of the ground state in position space, [53] andx A,B are the position operators before the displacements. We first consider the atomic stateρ A under one choice of the displacements. The reduced density matrix of the atoms can be obtained by integrating Ψ g Ψ * g over the coordinate of the field operatorX B , as done in Ref. [7]. The result has the same form as that of a thermal oscillator (see Appendix, also Ref. [54]), with unit mass and the effective oscillation frequency where we have set c = cos γ, s = sin γ, and cosh(βΩ) = 1 + 2ε 1 ε 2 [(ε 1 − ε 2 ) 2 c 2 s 2 ] −1 , with β = (k B T ) −1 and the Boltzmann constant k B . In the Fock basis of the thermal oscillator [54], the reduced density matrix of the atoms can be expressed aŝ whereĤ A = (P 2 A + Ω 2X2 A )/2 is the effective Hamiltonian of the thermal oscillator [7], with the eigenvectors |ψ n ≡ (â † Ω ) n |0 / √ n! and the eigenvalues p n ≡ ψ n |ρ A |ψ n . Here, the momentum operator is given byP for the annihilation operator of the atomic fluctuation δâ and that of the thermal oscillatorâ Ω . The position operator of the atomsX A also has a simple relationship with that of the thermal oscillator,X A = (δâ † + δâ)/ √ 2 ω = (â † Ω +â Ω )/ √ 2Ω. According to Ref. [18], the two displacements result in doubly degenerate and orthogonal ground states in the symmetrybroken (i.e., superradiant) phase, which in turn gives the two atomic statesρ ± A for each displacement. Obviously, we can diagonalize them in the two ortho-normalized Fock basis {|ψ ± n }, with ψ ± n |ψ ± n ′ = δ n,n ′ and ψ ± n |ψ ∓ n ′ = 0. The total atomic state is supposed to be an incoherent superposition ofρ ± A , i.e., ρ A = (ρ + A +ρ − A )/2. Note that the collective spin operators under the two displacements take the form andĴ z = (δâ † ± α s )(δâ ± α s ) − N/2, where the terms ∼ O(N 0 ) are neglectable in the thermodynamic limit. Using the selfconsistent condition δâ = 0 [18], it is easy to obtain the expectation values Ĵ x ± = ±α s N − α 2 s and Ĵ y ± = 0 for eachρ ± A . A 50:50 weighted average over Ĵ x ± gives Ĵ x = Ĵ y = 0 for the total density matrixρ A . As in the previous finite-N case, the squeezing parameter is given by ξ 2 = 4 Ĵ 2 y /N, with its explicit form where, in the the last step, we have dropped the intermediate quantities Ω and e βΩ (see Appendix). The reduced variance Ĵ 2 y ∝ (δâ † − δâ) 2 can also be obtained as previous work [18]. To obtain the QFI, one has to diagonalize the reduced density matrix as Eq. (8), and then calculate the QFI for eachρ ± A using Eq. (2). Since both of them are the same, we obtain the total QFI of the atoms which shows an exact relationship with the reduced variance, F A ξ 2 = Nµ 2 [53]. This finding can be used to verify that the enhanced QFI beyond the classical limit is induced by the squeezing, i.e., ξ 2 = ω 0 (ω 2 + ω 2 0 ) −1/2 < 1 and hence F A /N = ξ −2 > 1 at λ = λ cr .
For the field subsystem, one can obtain the reduced density matrixρ ± B for each displacement, similar to Eq. (8), but with different oscillation frequency Ω| c↔s , i.e., interchanging c and s in Eq. (7). Again, we assume that the total field state is a mixture ofρ ± B , which can be diagonalized in the Fock basis {|ψ ± n }. With the self-consistent condition δb = 0, it is easy to obtain X π/2 = P B / √ 2ω = 0 [18], whereP is the quadrature operator of the bosonic field, defined by Eq. (3). For eacĥ ρ ± B , we find that the reduced variances (∆X π/2 ) 2 ± = P 2 B ± /(2ω) take the same form (see Appendix), and therefore Again, we have removed the intermediate quantities Ω and e βΩ in the the last step. The QFI of the field mode depends on the matrix elements ψ ± m |Ĝ|ψ ± n , whereĜ =b †b = (δb † ∓ β s )(δb ∓ β s ) denotes the phase-shift generator and δb † ± δb ∝ (b † Ω ± b Ω ), as mentioned above. After some tedious calculations, we obtain the total QFI of the field mode (see Appendix) where β s is the order parameter, given by Eq. (5). In the normal phase, the first term of Eq. (13) dominates due to β s = 0. On the contrary, for the superradiant phase, the first term vanishes quickly and the second term becomes important due to the macroscopic occupation β 2 s ∼ O(N) → ∞, which gives a simple relation F B (∆X π/2 ) 2 ≈ β 2 s ≈n (see Appendix for the explicit form ofn).
Finally, let us investigate the scaling behaviors of the QFI of the atoms F A /N and that of the field mode F B /(4n) at λ ∼ λ cr . The critical exponents of a quantum phase transition are manifested in the behavior of the excitation energies [1]. For the Dicke model, it has been shown that the lower-branch excitation energy vanishes at λ = λ cr as ε 1 ∼ |λ − λ cr | 2ν , with the critical exponent ν = 1/4 [18]. Recently, Nataf et al. [33] have found that quantum fluctuations of the field ∆X 0 ∆X π/2 diverges as |λ − λ cr | −1/4 near the critical point. Here, we show that the QFI of the atoms F A /N is nonanalytic at λ = λ cr , since its first-order derivative diverges as ∂ λ (F A /N)| λ→λ cr ±0 ∼ |λ − λ cr | −1/2 . A similar result can be obtained for the field mode, F B /(4n), indicating that the QFIs of both subsystems are sensitive to the quantum criticality of the Dicke model, as one expects.

IV. CONCLUSION
In summary, we have investigated the quantum Fisher information of the field and that of the atoms in the ground state Dicke model. For finite and large enough N, we find that the QFI of each subsystem can beat the classical limit near the critical point λ cr , due to the appearance of a nonclassical squeezed state, as demonstrated numerically by the quasiprobability distribution ofρ A,B and the reduced quadrature variance below the classical limit. When the atom-field coupling enters the ultra-strong regime λ ≫ λ cr , we find the QFI of the bosonic field F B → 4n, while for the atoms F A → 0, sinceρ A,B at λ → ∞ is an incoherent mixture of two coherent states | ± α 0 and that of the atoms | j, ± j x , respectively. In the thermodynamic limit, we present analytical relations of the QFIs and the reduced variances for both subsystems, F A ξ 2 = Nµ 2 and F B (∆X π/2 ) 2 ≈n, which verify that the enhanced QFI near λ cr is induced by the squeezing. For each subsystem, we find that the first-order derivative of the QFI diverges as λ → λ cr ± 0, a sensitive probe of the superradiant quantum phase transition.