Spin squeezing and concurrence under Lee-Yang dephasing channels
SSpin squeezing and concurrence under Lee-Yang dephasing channels
Yuguo Su, Hongbin Liang, and Xiaoguang Wang ∗ Zhejiang Province Key Laboratory of Quantum Technology and Device,Department of Physics, Zhejiang University, Hangzhou, Zhejiang 310027, China
The Lee-Yang zeros are one-to-one mapping to zeros in the coherence of a probe spin coupled to a many-body system. Here, we study the spin squeezing under two different types of Lee-Yang dephasing channels inwhich the partition functions vanish at Lee-Yang zeros. Under the first type of the channels in which probes arecoupled to their own bath, we find that the performance of spin squeezing is improved and its maximum onlydepends on the initial state. Moreover, the centers of all the concurrence vanishing domains are correspondingto the Lee-Yang zeros. Under the second type of the channels in which probes are coupled to one bath together,the performance of spin squeezing is not improved, however, the concurrence shares almost the same propertiesunder both channels. These results provide new experimental possibilities in many-body physics and extend anew perspective of the relationship between the entanglement and spin squeezing in probes-bath systems.
I. INTRODUCTION
In 1952, when Lee and Yang studied the statistical theoryof equations of state and phase transitions, they proved thatthe partition functions of thermal systems exist zero roots,named Lee-Yang zeros, on the complex plane of fugacity ora magnetic field [1, 2]. They provided an insight into the theproblem of the Ising ferromagnet at arbitrary temperature inan arbitrary nonzero external fugacity or magnetic field. Asa cornerstone of statistical mechanics, they revealed the factthat under very general conditions, all the Lee-Yang zeros of ageneral Ising ferromagnet lie on the unit circle in the complexplane, which is the so-called unit-circle theorem [2]. Accord-ing to the celebrated theorem, under the thermodynamic limitcondition, the Lee-Yang zeros form a continuum ring in thecomplex plane [1]. Above the critical temperature, the contin-uum ring fractures and has a gap around the positive real axiswhich is a forbidden zone of the roots of the partition function[1], in other words, the free energy is analytic and there is nophase transition. Moreover, as the temperature decreases, thegap becomes narrowing and the two edge points approach thereal axis at the critical temperature [1]. Fisher [3] presentedthe concept, Yang-Lee edge singularities, which mean that thetwo edge points of the broken ring are singularity points [4].Liu [5] showed that the relationship between Lee-Yang zerosand the zeros in the coherence [6, 7] of a probe spin coupledto the many-body system is bijective. Furthermore, with thetemperature overtopping the critical point, sudden death andbirth of the coherence occurring at critical times are corre-sponding to the Yang-Lee singularities in the thermodynamiclimit.Because of the intrinsic difficulty that Lee-Yang zeroswould occur only at complex values of physical parameters,which are generally regarded as unphysical, physicists deemit hard to observe them. However, the Lee-Yang zeros havebeen observed in experiments [8], recently. The applicationsof the Lee-Yang theorem are manifold in general ferromag-netic Ising models of arbitrarily high spin [9–11], antiferro-magnetic Ising models [12], forecasting the large-deviation ∗ [email protected] statistics of the activity [13] as well as other striking typesof interactions [14–16].In the past decades, spin squeezing has attracted lots of at-tention [17–24]. It is well known that there are close relationsbetween entanglement and spin squeezing and many effortshave been devoted to unveiling it [25–30]. With wide mul-tipartite entanglement witnesses, spin squeezing is relativelyeasy to generate and measure experimentally [31–33]. Im-proving the precision of measurements is another importantapplication of spin squeezing. For example, spin squeezingplays an important role in Ramsey spectroscopy [18, 34–37],as well as in making high-precision atomic clocks [22, 38, 39]and gravitational-wave interferometers [40, 41], etc. B a t h 1
P r o b e 1
B a t h 2
P r o b e 2
B a t h N
P r ob e N ( a ) P r o b e 1 P r ob e N
B a t h
P r o b e 2 ( b ) Figure 1. Two different dephasing channels under a probe(s)-bathsystem. (a) N dephasing channels in which probes are only respec-tively coupled to their own bath. (b) The dephasing channel in which N probes are coupled to one bath together. The red circles markprobes and the blue circles mark bath spins. The coherence of a probe coupled to a many-body systemvanishes at times in one-to-one correspondence to the Lee-Yang zeros. In this work, we study the spin squeezing un-der two different types of Lee-Yang dephasing channels in a r X i v : . [ qu a n t - ph ] J un which the partition functions vanish at Lee-Yang zeros. Un-der the first type of the channels in which probes are coupledto their own bath [Fig. 1(a)], we calculate the coherence, therescaled concurrence and the spin-squeezing parameter underthe channel. We find that the performance of spin squeez-ing is improved and its maximum only depends on the initialstate. Moreover, the centers of all the concurrence vanish-ing domains are corresponding to the Lee-Yang zeros. Underthe second type of the channels in which probes coupled toone bath together [Fig. 1(b)], the performance of spin squeez-ing is not improved. However, the concurrence shares almostthe same properties under both channels. Furthermore, un-der the latter channel, the coherence could reaches the sta-ble value (cid:0) − cos N − θ (cid:1) / at times in correspondence to theLee-Yang zeros. These results provide new experimental pos-sibilities in many-body physics and extend a new perspectiveof the relationship between the entanglement and spin squeez-ing in probes-bath systems.This paper is organized as follows. In Sec. II, we intro-duce the Lee-Yang zeros and the dephasing channels. Theinitial state of the probe-bath systems and definitions of thespin-squeezing and the concurrence are given in Sec. III. InSec. IV, we introduce the two different types of Lee-Yang de-phasing channels and two spin-squeezing qubites as the initialstate. The coherence, the rescaled concurrence and the spin-squeezing parameter of multiporbes coupled to their own bathare given in Sec. V, and we discuss the influence of the bathspins number N b , the inverse temperature β , the probes num-ber N and the twisted angle θ on the them. In Sec. VI, wecalculate and analyse the coherence, the rescaled concurrenceand the spin-squeezing parameter of multiprobes coupled toone bath together. Finally, a summary is given in Sec. VII. II. LEE-YANG ZEROS AND DEPHASING CHANNELS
We focus on a general Ising model with ferromagnetic in-teractions under a magnetic field h . The Hamiltonian [5] is H ( h ) = − (cid:88) i,j λ ij s i s j − h (cid:88) i s i , (1)where the spins s i take values ± and the interactions λ ij ≥ .If the external magnetic field does not exist, all spins instate − or +1 is the ground state, which means it is com-pletely in order. Expanded by an N th order polynomial of z ≡ exp ( − βh ) , the partition function of N b spins at tem-perature T can be rewritten as Z ( β, h ) = Tr (cid:2) e − βH (cid:3) = e βN b h N b (cid:88) n =0 f n z n , (2)where β = 1 /T is the inverse temperature (Boltzmann andPlanck constants taken unity) and f n is the partition functionwith zero magnetic field under the condition that n spins are inthe state − (equivalently, we can also take the n spins in thestate +1 , since we know f n = f N b − n ). In principle, all phys-ical properties of an exotic system can be calculated from the partition function. Obviously, according to the notable unit-circle theorem, the N b zeros of the partition function, whichare located on the unit circle in the complex plane of z [2], canbe rewritten as z n ≡ exp ( i φ n ) with n = 1 , , . . . , N b . Sup-plied the Lee-Yang zeros as a priori knowledge, the partitionfunction can be expressed as Z ( β, h ) = f e βN b h N b (cid:89) n =1 ( z − z n ) , (3)by employing the polynomial factorization. Observing theequation, we can easily know that we can’t gain the z n or φ n straightway. Lemma: Under the periodic boundary condition, if z =exp ( i φ ) is a zero of the partition function Z ( β, h ) , its com-plex conjugate z ∗ = exp ( − i φ ) is also a zero.Proof: We first consider N b is even, since the periodic bound-ary condition f n = f N b − n , we gain N b (cid:88) n =0 f n z n = Nb − (cid:88) n =0 f n (cid:0) z n + z N b − n (cid:1) + f Nb z Nb = Nb − (cid:88) n =0 f n (cid:16) e n i φ n + e ( N b − n ) i φ n (cid:17) + f Nb e Nb i φ n = 0 . (4)Multiplying both sides of the above equation by the factor exp ( − N b i φ n ) , we have Nb − (cid:88) n =0 f n (cid:16) e ( N b − n )( − i φ n ) + e n ( − i φ n ) (cid:17) + f Nb e Nb ( − i φ n ) = 0 . (5)Therefore, z ∗ = exp ( − i φ n ) are the roots of the partitionfunction Z ( β, h ) .Then, we consider N b is odd, we have N b (cid:88) n =0 f n z n = Nb − (cid:88) n =0 f n (cid:0) z n + z N b − n (cid:1) = Nb − (cid:88) n =0 f n (cid:16) e n i φ n + e ( N b − n ) i φ n (cid:17) = 0 . (6)Employing the same skill, we can find that z ∗ = exp ( − i φ n ) are the roots of the partition function Z ( β, h ) . (cid:4) It explains the fact that distribution of the Lee-Yang zerosare symmetric about the real number axis. From the Lemma,we have the following proposition.
Proposition: Under the periodic boundary condition, thepartition function Z ( β, i x ) is a real and even function im-plying the relationship Z ∗ ( β, i x ) = Z ( β, i x ) = Z ( β, − i x ) ,where x is real. Proof:
From the Lemma, we know that N b (cid:89) n =1 (cid:0) a + b e i φ n (cid:1) = N b (cid:89) n =1 (cid:0) a + b e − i φ n (cid:1) , (7)where a and b are arbitrary complex numbers. We calculatethe ratio of them Z ∗ ( β, i x ) Z ( β, i x ) = e − i βN b x N b (cid:89) n =1 e i βx − e − i φ n e − i βx − e i φ n = N b (cid:89) n =1 − e − i βx − i φ n e − i βx − e i φ n = N b (cid:89) n =1 (cid:0) − e − i φ n (cid:1) . (8)From Eq. (7), we have Z ( β, i x ) Z ∗ ( β, i x ) = N b (cid:89) n =1 − e i βx + i φ n e i βx − e − i φ n = N b (cid:89) n =1 (cid:0) − e i φ n (cid:1) = N b (cid:89) n =1 (cid:0) − e − i φ n (cid:1) , (9)where a = 0 and b = − . Therefore, we find that Z ∗ ( β, i x ) Z ( β, i x ) = Z ( β, i x ) Z ∗ ( β, i x ) , which means Z ∗ ( β, i x ) = Z ( β, i x ) . Employing thesame skill, we can also see that Z ( β, − i x ) Z ( β, i x ) = Z ( β, i x ) Z ( β, − i x ) , and thepartition function Z ( β, i x ) is a real and even function, where x is real. (cid:4) Here, we introduce the dephasing channels. We write thetotal Hamiltonian of the system as follows, H = H ⊗ I × + ηH ⊗ σ z , (10)where H is the Hamiltonian of the environment, ηH ⊗ σ z is the Hamiltonian of the spin-environment interaction, η is acoupling constant, H = − (cid:80) j s j acts as the random field forthe probe spin, σ z = ( | (cid:105)(cid:104) | − | (cid:105)(cid:104) | ) is the Pauli matrix and I × is the identity operator.Therefore, we have the unitary matrix U ≡ e − i Ht = e − i H t e − i ησ z H t . (11)Employing the unitary transformation, we can gain the timeevolution density matrix of the system ρ ( t ) = U ( ρ B (0) ⊗ ρ P (0)) U † , (12)where ρ P (0) = ρ | (cid:105)(cid:104) | + ρ | (cid:105)(cid:104) | + ρ | (cid:105)(cid:104) | + ρ | (cid:105)(cid:104) | (13) and ρ B (0) are the initial density matrices of the spin and theenvironment, respectively.Hence, we can calculate the elements of the spin densitymatrix as follows, T ij = Tr B (cid:2) (cid:104) i | U ( ρ B (0) ⊗ ρ P (0)) U † | j (cid:105) (cid:3) , (14)where σ z | j (cid:105) = ( i ) j | j (cid:105) ( i, j = 0 , , and exp ( σ z ) = (cid:80) ∞ n =0 σ nz /n ! . Then we finde ± i ησ z H t | j (cid:105) = e ± ( i ) j +1 ηH t | j (cid:105) . (15)Employing Eq. (15), so we can rewrite Eq. (14), and get T = ρ Tr B [ e − i H t e − βH Z ( β, h ) e i H t ] = ρ . (16)Following this method, we gain the evolution of the reducedinitial density matrix ρ ( t ) = (cid:18) ρ µρ νρ ρ (cid:19) , (17)where the coefficients µ = Tr B (cid:20) e − βH Z ( β, h ) e − i ηH t (cid:21) , (18) ν = Tr B (cid:20) e − βH Z ( β, h ) e i ηH t (cid:21) . (19)Comparing with the definition of the dephasing channel, E (cid:18) ρ ρ ρ ρ (cid:19) = (cid:18) ρ (1 − p ) ρ (1 − p ) ρ ρ (cid:19) , (20)one can easily find that ρ ( t ) is a quantum dephasing channelwhen p = 1 − µ = 1 − ν , and the three Kraus operators aregiven by M = √ µI, M = (cid:112) − µ | (cid:105)(cid:104) | , M = (cid:112) − µ | (cid:105)(cid:104) | , (21)where I is the identity operator. In Heisenberg picture, a quan-tum channel with the Kraus operators is defined via the map E ( ρ ( t )) = (cid:88) i M i ρ ( t ) M † i , (22)and an expectation value of the operator O can be gained by (cid:104) O (cid:105) = Tr [ O E ( ρ ( t ))] = Tr (cid:2) E † ( O ) ρ ( t ) (cid:3) . (23) III. INITIAL STATE, AND DEFINITIONS OFSPIN-SQUEEZING AND CONCURRENCE
With so many benefits of spin squeezing, we are driven toapply spin squeezing to dephasing channels. We consider anensemble of N spin- / probes with ground state | (cid:105) and ex-cited state | (cid:105) . To take advantage of exchange symmetry, wechoose the one-axis twisted state | ψ (0) (cid:105) = e − i θJ x / | (cid:105) ⊗ N ≡ e − i θJ x / | (cid:105) , (24)where N is the number of the total qubits, the state is pre-pared by the one-axis twisting Hamiltonian H = χJ x , withthe coupling constant χ and θ = 2 χt being the twist angle.We set the mean spin of the initial state along the z direction,the two-qubit reduced density matrix becomes ρ P (0) = v + u ∗ y y y y u v − (25)in the basis {| (cid:105) , | (cid:105) , | (cid:105) , | (cid:105)} , where v ± = (1 ± (cid:104) σ z (cid:105) + (cid:104) σ z σ z (cid:105) ) / , (26) y = (cid:104) σ σ − (cid:105) , (27) u = (cid:104) σ − σ − (cid:105) . (28)For the one-axis twisted state, one could have [42] (cid:104) σ z (cid:105) = − cos N − (cid:18) θ (cid:19) , (29) (cid:104) σ z σ z (cid:105) = (cid:0) N − θ (cid:1) / , (30) (cid:104) σ σ − (cid:105) = 18 (cid:0) − cos N − θ (cid:1) , (31) (cid:104) σ − σ − (cid:105) = − (cid:0) − cos N − θ (cid:1) − i (cid:18) θ (cid:19) cos N − (cid:18) θ (cid:19) . (32)Then we discuss the spin-squeezing parameter defined byRef. [17] and we rewrite it as follows: ξ ≡ J ⊥ ) min N . (33)For our initial state, it can be simplify as ξ = 1 + 2 ( N −
1) ( (cid:104) σ σ − (cid:105) − |(cid:104) σ − σ − (cid:105)| ) . (34)The entanglement of formation is defined as follows Ref.[43], and the concurrence quantifying the entanglement of apair of spin- / is defined as C ( ρ ) = max { , λ − λ − λ − λ } , (35)where the λ i s are the square roots of the eigenvalues, in de-creasing order, of the non-Hermitian matrix R ≡ ρ ( t ) ( σ y ⊗ σ y ) ρ ∗ ( t ) ( σ y ⊗ σ y ) , (36)and ρ ∗ ( t ) denotes the complex conjugate of ρ ( t ) .From Ref. [23], we know that if ξ ≤ for even andodd states, then we gain the relationship between the spin-squeezing parameter and the rescaled concurrence: ξ =1 − ( N − C . To deliberate on the relationship, we take ξ (cid:48) ≡ − C r (37)for convenience, where C r = ( N − C ( ρ ( t )) is therescaled concurrence. We will examine the relationship un-der the Lee-Yang dephasing channels shown in Sec. IV. IV. TWO TYPES OF LEE-YANG DEPHASING CHANNELS
In this section, we introduce two different types of Lee-Yang dephasing channels shown in Fig. 1. As shown inFig. 1(a), each dephasing channel, in which probe is only cou-pled to its own bath, is independent. The dephasing channelhas N probes and each probe is equally coupled to all the N b bath spins [Fig. 1(b)]. A. Dephasing channel in which probe are coupled to their ownbath
We consider the dephasing channel in which probe is onlycoupled to its own bath and use two spin-squeezing probesspin- / coupled to the Ising system (bath). Then we can getthe total Hamiltonian H I = H ⊗ I × + ηH ⊗ σ z , (38)where η is a coupling constant, σ z = ( | (cid:105)(cid:104) | − | (cid:105)(cid:104) | ) is thePauli matrix. The probe spin lying on the bath is equally cou-pled to all the N b spins in the bath.Therefore, we have the unitary matrix and the time evolu-tion density matrix of the system U I ≡ e − i H I t = e − i H t e − i ησ z H t , (39) ρ I ( t ) = U I ( ρ P (0) ⊗ ρ B (0)) U † I , (40)where ρ P (0) = ρ | (cid:105)(cid:104) | + ρ | (cid:105)(cid:104) | + ρ | (cid:105)(cid:104) | + ρ | (cid:105)(cid:104) | (41)and ρ B (0) are the initial density matrices of the probes andthe bath, respectively.From Eqs. (14) and (15), we can calculate the elements ofthe probe spins density matrix and gain the evolution of thereduced initial density matrix ρ I ( t ) = E (cid:18) ρ ρ ρ ρ (cid:19) = (cid:18) ρ A + ρ A − ρ ρ (cid:19) (42)where A ± = Z ( β, h ± i ηt/β ) Z ( β, h )= e i N b ηt (cid:81) N b n =1 (cid:0) e − β ( h +2 i ηt/β ) − z n (cid:1)(cid:81) N b n =1 ( e − βh − z n ) (43)is the analogous partition function. From the Proposition, ifwe consider the system in a ferromagnetic Ising bath underzeros field ( h = 0 ), we can verify that the analogous partitionfunction A ( i x ) = e i βN b x (cid:81) N b n =1 (cid:0) e − i βx − e i φ n (cid:1)(cid:81) N b n =1 (1 − e i φ n ) (44)is a real even function, where x is real, implying the relation-ship A ∗ ( i x ) = A ( i x ) = A ( − i x ) . By this way, we denote A = A ( ± i ηt/β ) . Comparing with the definition of the de-phasing channel, E (cid:18) ρ ρ ρ ρ (cid:19) = (cid:18) ρ (1 − p ) ρ (1 − p ) ρ ρ (cid:19) , (45)one can easily find that ρ σ ( t ) is a quantum dephasing channel,where p = 1 − A , and the three Kraus operators are given by M = √ AI, M = √ − A | (cid:105)(cid:104) | , M = √ − A | (cid:105)(cid:104) | , (46)where I is the identity operator. B. Dephasing channel in which N probes are coupled to onebath together. We consider the dephasing channel in which multiprobesare coupled to one N b -spins bath together. From the previoussubsection, we can gain the total Hamiltonian and the unitarymatrix H II = H ⊗ I N × N + ηH ⊗ N (cid:88) k =1 σ k,z , (47) U II ≡ e − i H II t = e − i H t e − i η (cid:80) Nj = k σ k,z H t . (48)Here, we only consider the two probes case. The densitymatrix of the general initial state is given by ρ II (0) = (cid:88) i,j =0 ρ ij | i (cid:105) (cid:104) j | ⊗ ρ B (0) , (49)in the basis {| (cid:105) , | (cid:105) , | (cid:105) , | (cid:105)} . From Eqs. (12), (14)-(15) and (47)-(48), the reduced initial density matrix ρ (0) evolves to ρ II ( t ) = ρ Aρ Aρ A (cid:48) ρ Aρ ρ ρ Aρ Aρ ρ ρ Aρ A (cid:48) ρ Aρ Aρ ρ . (50)where A (cid:48) = A ( ± i ηt/β ) if we consider the system underzeros field ( h = 0 ). Because the relationship between A and A (cid:48) is uncertain, the evolution of a general initial state is not aquantum channel. However, if we choose a standard one-axistwisted state as Eq. (25), the evolution of the reduced initialstate is ρ II ( t ) = v + A (cid:48) u ∗ w y y w A (cid:48) u v − . (51)Therefore, we can find that ρ ( t ) is a quantum dephasing chan-nel and the Kraus operators are given by M = √ − A (cid:48) | (cid:105)(cid:104) | , M = √ − A (cid:48) | (cid:105)(cid:104) | ,M = √ A (cid:48) ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) , M = | (cid:105)(cid:104) | + | (cid:105)(cid:104) | . (52)Actually, if we consider the two-qubit reduced density matrixin multiprobes case, it will share the same channel. V. MULTIPROBES ONLY COUPLED TO THEIR OWNBATH
In this section, we calculate the coherence, the rescaledconcurrence and the spin-squeezing parameter of multiprobesonly respectively coupled to their own bath. And we discussthe influence of the bath spins number N b , the inverse temper-ature β , the probes number N and the twisted angle θ on thethem.We consider the standard one-axis twisted state as the initialstate under the dephasing channel. Therefore, we have thetotal Hamiltonian and the unitary matrix H III = N (cid:88) k =1 ( H ,k ⊗ I × + ηH ,k ⊗ σ k,z ) , (53) U III ≡ e − i H III t = N (cid:89) k =1 U k = N (cid:89) k =1 e − i H ,k t e − i ησ k,z H ,k t , (54)where H ,k and H ,k denote the Hamiltonian of the k thenvironment and the k th random field, respectively. FromEqs. (12), (14)-(15) and (53)-(54), the evolution of the reducedinitial state is ρ III ( t ) = v + A u ∗ w A y A y w A u v − , (55)and the coherence of the system is L ( t ) = A ( | u | + | u ∗ | + 2 | y | ) = 2 A ( | u | + y ) . (56)To give a specific impression about our idea, we elaborateon it by the one-dimensional (1D) Ising model with nearest-neighbor ferromagnetic coupling λ = 1 and the periodicboundary condition. Fig. 2 shows the Lee-Yang zeros andthe coherence of two spin-squeezing probe spins coupled totheir own bath at various parameters. At infinite temperature( β = 0 ), all the Lee-Yang zeros are degenerate at z n = − and we can easily find that the Lee-Yang zeros ( z n ) only de-pend on the particle number ( N b ) and the temperature ( β or /T ) . As shown in Figs. 2(a) and 2(b), one can find thatthe coherence decreases to zero and emerges N b peaks af-ter the first Lee-Yang singularity. With the bath size increas-ing toward the thermodynamic limit, the values of N b peaksdecrease dramatically. Obviously, the period of the coher-ence is T A = 2 π/ (4 η ) . From Eq. (44), supplied the uni-formly distributed Lee-Yang zeros ( φ n → (2 n − π/N b ,with β → ∞ ), the coherence is recovered when the timesatisfies t = (2 n − T A / (2 N b ) , where n = 1 , , , . . . [Fig. 2(c)]. As shown in Fig. 2(d), the probes number N andthe twist angle θ only impact on the amplitude of the coher-ence. Furthermore, one can find that times in correspondenceto the Lee-Yang zeros are also the zeros of the coherence.Now, we calculate the entanglement of the probes-bath sys-tem. As a prior knowledge, from Eqs. (26)-(32), we know √ v + v − ≥ | u | , v ± and y = w > are real. Employing L A t L ( t ) , A
20 30 40 50 60 70 80 t L ( t ) N b = β = = θ = π ( a ) L A t L ( t ) , A
20 21 22 23 24 t L ( t ) N b = β = = θ = π ⨯ - ( b ) L A t L ( t ) , A N b =
100 , β =
10, N = θ = π ( c ) L A t L ( t ) , A N b =
100 , β =
10, N = θ = π ( d ) Figure 2. Relationship between the Lee-Yang zeros and the coher-ence of two spin-squeezing probe spins coupled to their own bath.(a), (b) Coherence (blue line) and the analogous partition function A (red dot dash line) for a 1D Ising model with the number of spins N b = 10 , , respectively, for the inverse temperature β = 0 . ,the number of the probes N = 3 and the twist angle θ = π/ . Thecoherence and the function A with N b = 100 and β = 10 for (c) N = 3 , θ = π/ and (d) N = 5 , θ = π . The insets in (a) and (b)zoom into the coherence zero. The blue lines mark the coherencesand the red dot dash lines mark the functions A . The probes-bathcoupling constant is η = 0 . . Eqs. (36) and (55), we immediately get the square roots ofeigenvalues λ , = √ v + v − ± A | u | , (57) λ , = (cid:0) ± A (cid:1) y. (58)From Eq. (35), we obtain the evolution of the concurrence as C ( ρ ( t )) = 2 max (cid:8) , A | u | − y, A y − √ v + v − (cid:9) . (59)From the result we have gained, C r = max (cid:8) , A C r (0) + 2 ( N − (cid:0) A − (cid:1) y (cid:9) , (60)where C r (0) = 2 ( N − max { , | u | − y } is the initial con-currence.To illustrate the above result, we test the one-dimensional(1D) Ising model with nearest-neighbor ferromagnetic cou-pling λ = 1 and the periodic boundary condition with ferro-magnetic Ising bath under zeros field ( h = 0 ).Fig. 3 shows the rescaled concurrence of the two spin-squeezing probes coupled to their own bath with probes-bathcoupling η = 0 . , N b = 100 bath spins and the twist an-gle θ = π/ at various temperatures β and the number ofprobes N . As seen in Fig. 3(a), under the thermodynamic C r A t C r , A t C r , A N b = β = = θ = π ( a ) C r A t C r , A N b = β =
10, N = θ = π ( b ) C r A t C r , A N b = β =
10, N = θ = π ( c ) C
10 20 30 40 5010 - - - - N C m a x β = , t = π η N b , θ = π — -( N - ) ( d ) Figure 3. Relationship between the Lee-Yang zeros and the rescaledconcurrence of the two spin-squeezing probes coupled to their ownbath with various inverse temperatures β and the number of probes N . The rescaled concurrence (green line) and the analog partitionfunction A (red dot dash line) for (a) β = 0 . , N = 3 , (b) β = 10 , N = 3 and (c) β = 10 , N = 6 . (d) Maximum original concurrence(black dots) and the fitting function (red line) at the inverse tempera-ture β = 10 . The probes-bath coupling is η = 0 . , the number ofthe bath spins N b = 100 and the twist angle θ = π/ . limit, the rescaled concurrence decreases rapidly to zero withproper temperature β . However, with low temperature, therescaled concurrence emerges peaks at t = nT A /N b , where n = 1 , , , . . . [Fig. 3(b)]. The probes number N and thetwist angle θ only impact on the amplitude of the rescaledconcurrence [Fig. 3(c)]. Times in correspondence to the Lee-Yang zeros are the centers of all the vanishing domains ofthe rescaled concurrence at low temperature. As shown inFig. 3(d), given the twist angle θ , the maximum original con-currence is C max ( ρ ) = exp [ α ( N − , where α is a constantonly depended on θ . With θ = π/ and low temperature,the maximum original concurrence is C max ( ρ ) = 2 − ( N − attimes satisfying t = nT A /N b .From Eqs. (22) and (46), the evolution of the matrix O un-der the dephasing channel is E ( O ) = E † ( O ) = (cid:18) O AO AO O (cid:19) , (61)and we find that E † ( σ z ) = σ z (62) E † ( σ i ) = Aσ i for i = x, y. (63)From Eqs. (23) and (63), one have (cid:104) σ σ − (cid:105) = A (cid:104) σ σ − (cid:105) , (64) (cid:104) σ − σ − (cid:105) = A (cid:104) σ − σ − (cid:105) , (65)where (cid:104) O (cid:105) = (cid:104) ψ (0) | O | ψ (0) (cid:105) .Since each probe is only coupled to its own bath, the in-dependent and identical dephasing channels act separately oneach probe spin. Consequently, the spin-squeezing parametersare obtained as ξ = 1 − N − A ( | u | − y ) = 1 − A C r (0) , (66) ξ (cid:48) = 1 − max (cid:8) , A C r (0) + 2 ( N − (cid:0) A − (cid:1) y (cid:9) . (67)Therefore, under this dephasing channel, the spin-squeezingparameter is unbound by the relationship ξ = 1 − C r and theperformance of spin squeezing is improved by ∆ ξ ≡ ξ (cid:48) − ξ = 2 ( N − min (cid:8) A ( | u | − y ) , (cid:0) − A (cid:1) y (cid:9) . (68)The maximum of the improvement ∆ ξ max = 2 ( N − (cid:18) − y | u | (cid:19) y (69)only depends on its initial state. C r ξ ξ '2 t C r , ξ , ξ ' t C r , ξ , ξ ' N b = β = = θ = π ( a ) C r ξ ξ '2 t C r , ξ , ξ ' N b = β =
10, N = θ = π ( b ) Figure 4. Relationship between the rescaled concurrence and thespin-squeezing parameters of the two spin-squeezing probes coupledto their own bath with various inverse temperatures β and the numberof probes N . The rescaled concurrence (green line) and the spin-squeezing parameters ξ (red line), ξ (cid:48) (black line) for (a) β = 0 . , N = 3 , (b) β = 10 , N = 5 . The inset in (a) zooms into the initialtime. The probes-bath coupling is η = 0 . , the number of the bathspins N b = 100 and the twist angle θ = π/ . Fig. 4 shows the rescaled concurrence and the spin-squeezing parameters of the two spin-squeezing probes cou-pled to their own bath with probes-bath coupling η = 0 . , N b = 100 bath spins and the twist angle θ = π/ at vari-ous temperatures β and the number of probes N . As seen inFig. 4(a), the spin-squeezing parameter ξ is always smallerthen the corresponding ξ (cid:48) , which means it does not hold therelationship ξ = 1 − C r under decoherence. Moreover, withtemperature decreasing, the spin-squeezing parameter is re-covered periodically at time satisfying ( T A /N b ) [Fig. 4(b)]. VI. MULTIPROBES COUPLED TO ONE BATHTOGETHER
In this section, we obtain and analyse the coherence, therescaled concurrence and the spin-squeezing parameter of one probe coupled to a many-body system. We compare them ofboth systems we have mentioned.To take advantage of exchange symmetry, our initial state isone-axis twisted state shown as Eq. (25). From the dephasingchannel we showed in Sec. IV, one can gain the coherence L ( t ) = 2 ( | A (cid:48) u | + y ) . (70)Employing Eqs. (36) and (51), we obtain the square rootsof eigenvalues as follows: λ , = √ v + v − ± | A (cid:48) u | , (71) λ = 2 y, (72) λ = 0 . (73)From Eqs. (35) and (III), we obtain the evolution of therescaled concurrence as C r = 2 ( N − max (cid:8) , | A (cid:48) u | − y, y − √ v + v − (cid:9) = max { , | A (cid:48) | C r (0) + 2 ( N −
1) ( | A (cid:48) | − y } , (74)where C r (0) = 2 ( N − max { , | u | − y } is the initial con-currence.From Eqs. (22), (23) and (52), the evolution of the matrix O under the dephasing channel is E ( O ) = E † ( O ) = O A (cid:48) O O O O O A (cid:48) O O , (75)and we find that (cid:104) σ σ − (cid:105) = (cid:104) σ σ − (cid:105) , (76) (cid:104) σ − σ − (cid:105) = A (cid:48) (cid:104) σ − σ − (cid:105) . (77)Then we obtain the evolution of the spin-squeezing parameter ξ = ξ (cid:48) = 1 − C r = 1 − N −
1) ( | A (cid:48) u | − y ) . (78)That means decoherence does not destroy the relationship ξ + C r = 1 during the whole evolution. Analytical resultsfor time-evolution of all relevant coherences, rescaled concur-rences, expectations and spin-squeezing parameters, chosenone-axis twisted state as the initial state, are given in Table. I.We show our results by the one-dimensional (1D) Isingmodel with nearest-neighbor ferromagnetic coupling λ = 1 and the periodic boundary condition. Fig. 5 shows the co-herence, the rescaled concurrence and the spin-squeezing pa-rameter of two spin-squeezing probes coupled to one bathtogether. As shown in Figs. 5(a) and 5(b), with appropri-ate temperature, one can find that the coherence is asymp-totic to a stable value (cid:0) − cos N − θ (cid:1) / because the analo-gous partition function A (cid:48) becomes almost zero after the firstLee-Yang singularity. With the bath size increasing towardthe thermodynamic limit, one can gain the stable minimumvalue of the coherence in almost the whole time domain. FromEq. (44), when temperature is low enough ( β → ∞ ), the co-herence is recovered from the minimum value when the time Table I. Analytical results for time-evolution of all relevant coherences, rescaled concurrences, expectations and spin-squeezing parameters,chosen the one-axis twisted state as the initial state.Dephasing channels Dephasing channelin which multiprobes are coupled to their own bath in which multiprobes are coupled to one bath together L ( t ) 2 A ( | u | + y ) 2 ( | A (cid:48) u | + y ) Cr max (cid:8) , A C r (0) + 2 ( N − (cid:0) A − (cid:1) y (cid:9) max { , | A (cid:48) | C r (0) + 2 ( N −
1) ( | A (cid:48) | − y }(cid:104) σ z σ z (cid:105) (cid:104) σ z σ z (cid:105) (cid:104) σ z σ z (cid:105) (cid:104) σ σ − (cid:105) (cid:104) σ σ − (cid:105) (cid:104) σ σ − (cid:105) (cid:104) σ − σ − (cid:105) A (cid:104) σ − σ − (cid:105) A (cid:48) (cid:104) σ − σ − (cid:105) ξ − A C r (0) 1 − C r L2y A ' t L ( t ) , A ' N b = β = = θ = π ( a ) L2y A ' t L ( t ) , A ' N b = β = = θ = π L ( t ) + ⨯ - - ⨯ - t ( b ) L2y A ' - - - t L ( t ) , A ' N b = β =
10, N = θ = π ( c ) C r A ' t C r , A ' t C r , A ' N b = β = = θ = π ( d ) C r A ' - - - t C r , A ' N b = β =
10, N = θ = π ( e ) C r ξ t C r , ξ t C r , ξ N b = β = = θ = π ( f ) Figure 5. Coherence, rescaled concurrence and spin-squeezing pa-rameters of two spin-squeezing probes coupled to one bath together.The coherences (blue line) and the analogous partition functions A (cid:48) (red dot dash line) with the twist angle θ = π for (a) N b = 10 , β = 0 . , (b) N b = 100 , β = 0 . and (c) N b = 100 , β = 10 .The rescaled concurrences (green line) and the analogous partitionfunctions A (cid:48) (red dot dash line) with N b = 100 and θ = π/ for(d) β = 0 . and (e) β = 10 . (f) The rescaled concurrences (greenline) and the spin-squeezing parameters ξ (red line) for N b = 100 , β = 0 . and θ = π/ . The insets of (b) and (f) zoom into the coher-ence zero and the initial time, respectively. The probes-bath couplingis η = 0 . and the number of probes N = 3 . satisfies t = (2 n − T A (cid:48) / (2 N b ) , where T A (cid:48) = T A / and n = 1 , , , . . . [Fig. 5(c)]. As seen in Figs. 5(d) and 5(e), therescaled concurrence shares the same properties with the cor-respondence in Sec. V, except for the double period. Times incorrespondence to the Lee-Yang zeros are the centers of all thevanishing domains of the rescaled concurrence at low temper-ature. As shown in Fig. 5(f), one can find that the relationship ξ = 1 − C r still is preserved under decoherence. VII. CONCLUSION AND DISCUSSION
In conclusion, we first introduce the Lee-Yang zeros and thedephasing channels. Having introduced them, we propose twodifferent types of Lee-Yang dephasing channels: (a) probesare coupled to their own bath and (b) probes are coupled to onebath together. Under the Lee-Yang dephasing channels, weobtain the coherence, the concurrence and spin-squeezing pa-rameter, with nearest-neighbor ferromagnetic coupling λ = 1 ,and zeros field ( h = 0 ) and the periodic boundary condition.In the first place, one can find that the coherence decreases tozero (minimum, under the latter channel) and emerges certainpeaks after the first Lee-Yang singularity, of which values de-crease dramatically with the bath size increasing toward thethermodynamic limit. The coherence can be recovered at cer-tain discrete times if the temperature is low enough. Further-more, one can find that times in correspondence to the Lee-Yang zeros are also the zeros of the coherence. Secondly, theconcurrence shares almost the same properties in both chan-nels and the centers of its vanishing domains are correspond-ing to the Lee-Yang zeros. Besides, at the given the twist an-gle, the maximum original concurrence depends on the num-ber of probes. Finally, we find that the performance of spinsqueezing is improved and its maximum only depends on theinitial state under the first dephasing channel. However, thecorresponding performance is unimproved under the seconddephasing channel.All we have discussed above are in the ferromagnetic Isingmodels. For other systems (e.g., antiferromagnetic Ising mod-els), the Lee-Yang zeros may not lie on a unit circle. How-ever, according to Eq. (43), one can employ an external field h and gain all the zeros of modulus exp ( − βh ) . Becausethe zeros of coherence and the centers of all the concurrencevanishing domains are times in correspondence to the Lee-Yang zeros as the temperature approaches zero, one can gainthe information about the corresponding Lee-Yang zeros bymeasuring the quantum coherence and concurrence. With theLee-Yang zeros determined, the partition function of an in-triguing many-body system can be rebuilt and one can obtainall the properties of the system. Since all physical propertiescan be acquired, our results provide a new method to inves-tigate in many-body physics and extend a new perspective ofthe relationship between the entanglement and spin squeezing in probes-bath systems. VIII. ACKNOWLEDGMENTS
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