PHYSICAL REVIEW A 97, 023832 (2018)
Nonclassical states of light with a smooth P function
François Damanet,1,2 Jonas Kübler,3 John Martin,2 and Daniel Braun3 1Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom 2Institut de Physique Nucléaire, Atomique et de Spectroscopie, CESAM, Université de Liège, Bâtiment B15, B-4000 Liège, Belgium 3Institut für theoretische Physik, Universität Tübingen, 72076 Tübingen, Germany
(Received 12 December 2017; published 21 February 2018)
There is a common understanding in quantum optics that nonclassical states of light are states that do not have a positive semidefinite and sufficiently regular Glauber-Sudarshan P function. Almost all known nonclassical states have P functions that are highly irregular, which makes working with them difficult and direct experimental reconstruction impossible. Here we introduce classes of nonclassical states with regular, non-positive-definite P functions. They are constructed by “puncturing” regular smooth positive P functions with negative Dirac-δ peaks or other sufficiently narrow smooth negative functions. We determine the parameter ranges for which such punctures are possible without losing the positivity of the state, the regimes yielding antibunching of light, and the expressions of the Wigner functions for all investigated punctured states. Finally, we propose some possible experimental realizations of such states.
DOI: 10.1103/PhysRevA.97.023832
I. INTRODUCTION by Sudarshan [17] that in fact all possible quantum states ρ of the light field (quantum or classical) can be represented Creating and characterizing nonclassical states of light by (1). This is possible due to the fact that coherent states are are two tasks of substantial interest in a wide range of overcomplete. The price to pay, however, is that in general applications [1,2], such as sub-shot-noise measurements [3–5], P (α) is not a smooth function, but a functional. Obviously, high resolution imaging [6–9], light source and detector cal- indeed, even a coherent state |α is already represented by a ibration [10–12], quantum cryptography [13,14], or quantum 0 functional, namely a Dirac-δ peak, P (α) = δ(α − α ). Such information processing [15]. 0 a level of singularity represents the “boundary of acceptable In quantum optics, there is widespread agreement that the irregular behavior” of the P function for a classical state. Much most classical pure states of light are coherent states |α . These worse singularities arise for example from simple Fock states are eigenstates with complex eigenvalue α of the annihilation † |n , i.e., energy eigenstates of the mode, a a|n =n|n ,for operator a of the mode of the light field under consideration, which the P function is given by the nth derivative of a δ a|α =α|α , and are characterized by the minimal uncertainty √ function [18]. product of the quadratures X = (a + a†)/ 2 and Y = (a − √ It would be desirable that one could directly reconstruct a†)/(i 2). In the case of a mechanical harmonic oscillator the P function and in this way show the nonclassicality of the quadratures correspond simply to phase-space coordinates states that are genuinely quantum. However, in the case of x and p in suitable units. Hence a coherent state resembles highly nonregular P functions this is impossible. One way most closely the notion of a classical phase-space point located out is to apply filters that do not enhance the negativity of at the mean position of the quadratures. The label α ∈ C √ the characteristic function, i.e., the Fourier transform of P (α) determines these mean values as α|X|α = 2Re(α) and √ [19–22]. α|Y |α = 2Im(α). Coherent states of light, which can be Also for the ease of theoretical work one would like to produced by lasers operated far above the threshold [16], have classes of P functions that are regular and smooth, but retain the property of minimal uncertainty XY = 1/2, represent genuine quantumness due to the fact that they become = 2 − 2 where X X X is the standard deviation, under negative [23]. One class of such states are single-photon added the dynamics of the harmonic oscillator underlying each † thermal states [20], i.e., states of the form ρ = N a ρT a, where mode of the electromagnetic field. Mixing classically coherent ρT represents a thermal state and N is a normalization factor. states (i.e., choosing randomly such states according to a The P function of this state is given by the product of a linear classical probability distribution) should not increase their function of |α|2 and the Gaussian of the thermal state [see Eq. quantumness. Hence (possibly mixed) quantum states given (4.36) in [18]]. But as Agarwal writes (p. 84 in [18]), “Here we by a density operator ρ as perhaps have the unique case where P (α) exists but is negative. Most known cases of nonclassical P involve P functions which ρ = P (α)|α α| d2α (1) do not even exist.” In this paper, we introduce whole classes of such genuinely with a positive P function are considered “classical states” in quantum states with nevertheless smooth P functions (i.e., quantum optics. While it may appear that (1) describes only exhibiting at most singularities in the form of Dirac δ func- states that are “diagonal” in a coherent state basis, it was shown tions). The recipe for doing so is very simple: start with a
2469-9926/2018/97(2)/023832(12) 023832-1 ©2018 American Physical Society DAMANET, KÜBLER, MARTIN, AND BRAUN PHYSICAL REVIEW A 97, 023832 (2018) classical quantum state with a smooth positive P function and represent coherent states, we consider in the following sections then superpose some narrow, negative peaks. Altogether, one Dirac δ(α) functions as worst singularities for the punctures must guarantee, of course, that the density matrix ρ remains πi (α). This is the also the simplest case, as it does not contain a semidefinite positive operator. Finding parameter regimes any further free parameter. where this is the case constitutes the main technical difficulty for constructing such states. Obviously, this general recipe B. Conditions for (semi)positive definiteness allows for an infinity of possibilities, starting from the choice For a given choice of the functions πi (α) for the punctures, of the smooth original P function, over the number and form ∈ of the negative peaks superposed, to their amplitudes and positivity of the state will depend on the position of the αi C and the weights wi . Our task is hence to find the allowed positions. Westart with the simplest situation of the rotationally { } symmetric Gaussian P function centered at |α|=0ofa parameter ranges for wi ,αi such that ρ 0, i.e., ρ is a positive semidefinite operator. The normalization factor N thermal state, “punctured” with a single negative δ peak whose 2 = position and amplitude we can vary. Only the radial position is ensures that P (α)d α 1, i.e., − relevant here, and we have herewith already three parameters, N 1 exemplifying the wide range of tunability of punctured states. N = 1 − wi . (4) The paper is organized as follows. In Sec. II, we introduce i=1 the “punctured states,” their general properties, the conditions N for positive definiteness of their density-matrix representation, Since does not influence the positivity, we will often skip and the conditions yielding nonclassical states. In Sec. III, it. we investigate the particular simple case of a thermal state It is clear that in general the condition for any single punctured by a single δ peak. In Sec. IV, we study the parameter will depend on the values of all the others. This can case of a squeezed thermal state punctured by a δ peak. In be seen immediately from the general necessary and sufficient condition of positive semidefiniteness, ψ|ρ|ψ 0 ∀|ψ ∈ Sec. V, we generalize our results by considering a thermal state H ≡ 2 punctured by a Gaussian peak of finite width. In each section, L (C), which for the state (1) with a P function of the we provide the expression of the Wigner function, calculate the form (2) reads second-order correlation function and identify the regimes of P (α) − w π (α − α ) |ψ|α |2d2α 0 ∀|ψ ∈H . parameters yielding antibunching of light, i.e., sub-Poissonian cl i i i statistics of the photon counting. In Sec. VI, we discuss some i possibilities of experimentally creating some of these states. (5) Finally, in Sec. VII, we give a conclusion of our work. To go beyond this general criterion, we first review briefly a few necessary and sufficient conditions for positivity of a linear II. DEFINITION, GENERAL PROPERTIES, POSITIVITY, operator. AND NONCLASSICALITY CONDITIONS Necessary and sufficient condition (NSC). A necessary and sufficient condition (NSC) for a Hermitian matrix ρ to A. General punctured states be semipositive definite (ρ 0) is that all its eigenvalues are Let ρcl represent a classical state, that is a state as defined non-negative. While the eigenvalues of an infinite-dimensional in (1) with a smooth non-negative P function Pcl(α) 0. We matrix are generally not readily accessible, the condition define a “punctured ρcl state” as a state (1) with a P function nevertheless allows for a numerical approach. Indeed, the of the form density operator (1) can be expressed in the Fock state basis N {|n : n ∈ N0} using the expansion of coherent states = N − − P (α) Pcl(α) wi πi (α αi ) , (2) ∞ n − |α|2 α i=1 |α =e 2 √ |n . (6) n=0 n! where N is a normalization constant, the wi are positive coefficients representing the weights of the different punctures Then, by introducing a cutoff dimension in Hilbert space of = i (i 1,... ,N), and πi (α) are positive functions on the sufficiently high excitation nmax, a finite-dimensional matrix complex plane that determine their shape. We take them as representation can be obtained and one can check numerically normalized according to that, when increasing the cutoff, the lowest eigenvalue con- verges to a value that is sufficiently far from zero for judging 2 πi (α)d α = 1, ∀i = 1,... ,N. (3) the positivity of the state. Necessary condition (NC). Since ρ 0 ⇔ψ|ρ|ψ 0 If the weights are chosen such that the P function (2) for all states |ψ , a necessary condition (NC)forρ 0 is that becomes negative for some α and that the resulting state ρ all diagonal elements ρφφ ≡φ|ρ|φ of the density matrix are is a positive semidefinite operator with unit trace, it follows non-negative for a given class of states |φ ∈K ⊂ H, where immediately that the state is nonclassical. Furthermore, the K is a subset of H, i.e., P function remains “well behaved” (i.e., sufficiently regular) ρφφ ≡φ|ρ|φ 0, ∀|φ ∈K ⇐ ρ 0. (7) if πi (α) are well behaved, as we assumed Pcl(α) smooth. In agreement with the common understanding that positive Sufficient condition (SC). A sufficient condition (SC)for P functions corresponding to simple Dirac-δ peaks are still positivity can be found from Gershgorin’s Circle Theorem. accepted in the class of well-behaved functions because they Consider first the case of finite-dimensional matrices A.The
023832-2 NONCLASSICAL STATES OF LIGHT WITH A SMOOTH … PHYSICAL REVIEW A 97, 023832 (2018) theorem then says [24] “Every eigenvalue of a complex square where · =Tr( · ρ), or, equivalently, on the Mandel QM matrix A of coefficients {aij } lies within at least one of parameter [33] the Gershgorin disks D(a ,R ) centered at a with radii ii i ii 2 − = | | = | | (n) n (2) Ri j =i aij or Ri j =i aji .” An interpretation of this QM = =n (g − 1), (11) theorem is that if the off-diagonal elements of A have small n norms compared to its diagonal elements, its eigenvalues where n = a†a. A state is said to be nonclassical if g(2) < 1 cannot be far from the values of the diagonal elements. A (or, equivalently, QM < 0), which corresponds to antibunching matrix is called (strictly) diagonally dominant if |aii| Ri | | (sub-Poissonian statistics of photon counting that cannot be ( aii >Ri ) for all i. This leads to the following sufficient observed for mixtures of coherent states). condition: a Hermitian (strictly) diagonally dominant matrix It is interesting to note the connection between classicality with real non-negative diagonal entries is positive semidefinite in terms of the P function and in terms of antibunching, pointed (definite). Gershgorin’s Circle Theorem has been generalized out by Kimble et al. [34] and generalized by Vogel to space- to the case of infinite-dimensional matrices in [25], where a and time-dependent correlations [35]: g(2) < 1 can exist if proof was given for matrices of the space L1. The proof for the 2 the P function is negative somewhere. Conversely, if the P space L in which our operators live can be done analogously function of a state is a valid probability density, then there is and leads to the same result, namely that the eigenvalues lie in no antibunching, i.e., g(2) 1. ∪∞ D(a ,R ) (just as in the case of a finite space). Hence, by i=1 ii i Proof. This statement can be proved using Jensen’s inequal- expressing the density matrix in the Fock state basis, we obtain ity. We use Eq. (5’) in the original paper [36]. Let P (α)bea the sufficient condition (SC) valid probability density defined over the complex numbers. − ∀| ⇒ = 2π iφ ρnn Rn 0, n ρ 0, (8) Then Pr (r) 0 rP(re ) dφ defines a valid probability density over the real numbers r ∈ [0,∞). Using Eq. (1), the where ρ ≡n|ρ|n and R = |ρ |. nn n m =n nm second-order correlation function (10) can be written as ∞ |α|4P (α) d2α r4P (r) dr C. Conditions for nonclassicality (2) = = 0 r g ∞ . (12) | |2 2 2 2 2 There exist various nonclassicality criteria for radiation α P (α) d α 0 r Pr (r) dr fields. In this work, we shall consider criteria based on the = = 2 negativity of the P function, the negativity of the Wigner Defining the convex functions f (r) ϕ(r) r , we can function and the existence of antibunching [1]. rewrite the above as ∞ Negativity of the P function. A criterion for a state to be ϕ(f (r))P (r)dr (2) 0 r nonclassical is that its P function takes negative values, so g = ∞ . (13) ϕ f (r) P (r) dr that it can no longer be interpreted as a probability distribution 0 r in phase space. For example, all physical δ-punctured states The condition g(2) 1 is then equivalent to are nonclassical according to this criterion. Other punctured ∞ ∞ states, however, could have a non-negative P function in the ϕ(f (r))Pr (r)dr ϕ f (r) Pr (r) dr , (14) whole complex plane, so that general punctured states are not 0 0 necessarily nonclassical. which is true by Jensen’s inequality. Negativity of the Wigner function. Negativity of the Wigner As a conclusion, for radiation field states, both the negativity function is a widely used criterion for nonclassicality. This is of the Wigner function and antibunching imply that the P largely due to the fact that the Wigner function can be directly function is not a valid probability density, while the reverse W α measured (see, e.g. [26–31]). Since the Wigner function ( ) is not necessarily true. is the convolution of the P function with the vacuum state, i.e., In the following sections, we investigate the cases of various 2 − | − |2 punctured states. In each case, we apply the conditions of W(α) = P (β) e 2 α β d2β, (9) π positivity of and nonclassicality of ρ to determine the physical nonclassical states. For the sake of simplicity, we restrict the existence of negative values for the Wigner function implies ourselves in the remainder of the paper to a single puncture the existence of negative values for the P function, but the by setting N = 1inEq.(2). reverse is not true. Thus the criterion based on negative values of the Wigner function detects fewer nonclassical states than the previous criterion. More specifically, it does not III. δ-PUNCTURED THERMAL STATES detect states with partly negative P function but everywhere We define a δ-punctured thermal state as a state determined positive Wigner function. This explains why the definitions through (1) and (2) with of punctured states based on the puncturing of P functions rather than Wigner functions is more appropriate. Moreover, δ e−|α|2/n¯ P (α) = , puncturing of a Wigner function would be forbidden as Wigner cl πn¯ functions are always bounded (see, e.g., p. 73 in [32]). (15) Antibunching. Another standard criterion of nonclassicality π(α) = δ(α), is based on the second-order correlation function − where n¯ = (ehω/k¯ B T − 1) 1 is the average number of thermal a†a†aa excitations at temperature T of the radiation field with fre- g(2) = , (10) a†a 2 quency ω.
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A. Conditions for (semi)positive definiteness Necessary condition. A necessary condition for the positiv- ity of ρ can be obtained from the condition (7) with K the set of coherent states. It yields the following upper bound for the puncture weight w1: −| |2 e α1 /n¯ w ≡ wδ,NC. (16) 1 n¯ + 1 max Proof.Let|γ (γ ∈ C) denote a coherent state. The density- matrix element ργγ ≡γ |ρ|γ is then given by | |2 1 − α −|α−γ |2 2 −|α −γ |2 ρ = N e n¯ d α − w e 1 , (17) γγ πn¯ 1
2 −|α−γ |2 where we used |γ |α | = e . The condition that ργγ must be non-negative for all γ reads δ,NC |α|2 FIG. 1. Plot of the upper bounds wmax [Eq. (16), solid curves] − −|α−γ |2 2 SC 1 e n¯ d α wδ, wδ w ∀ ∈ and max [Eq. (22), dashed curves], and of max,the 1 which w1 −| − |2 γ C. (18) πn¯ e α1 γ cancels the smallest eigenvalue of the density matrix, as a function of |α | for different n¯ (green squares: n¯ = 0.1; blue dots: n¯ = 0.5; After integration, this yields the condition 1 orange triangles: n¯ = 1; purple diamonds: n¯ = 2). The eigenvalues − 1 |γ |2+|α −γ |2 = e ( n¯+1 ) 1 are computed in a Fock space of dimension nmax 15. Inset: conver- w ∀γ ∈ C. (19) gence of wδ (green squares) with increasing cutoff dimension n 1 n¯ + 1 max max (nmax = 2, 3, 4, and 15 from top to bottom) for n¯ = 0.1. The minimum over γ of the right-hand side (RHS) term = n¯+1 is attained for γ ( n¯ )α1 and directly leads to the condi- δ,NC tion (16). In Fig. 1, we display both wmax and the value of w1 that Sufficient condition. In the Fock state basis, the Gershgorin cancels the smallest eigenvalue of ρ. In our computations, the disks D(ρnn,Rn) have center ρnn ≡n|ρ|n and radius Rn density matrix is expressed in the truncated Fock state basis given by {|0 ,|1 ,... ,|nmax } with a sufficiently large cutoff nmax in n 2n order to ensure stable numerical results. An analytical proof n¯ −| |2 |α1| = − α1 of the tightness for a puncture at α = 0 will be provided in ρnn + w1e , (20) 1 (n¯ + 1)n 1 n! the context of Gaussian punctures (Sec. VA). Note that a δ δ,NC n +∞ m puncture at α = 0 with maximum weight w corresponds to 2 |α | |α | max −|α1| 1 1 Rn = w1e √ √ . (21) the complete removal of the ground state of the oscillator (“the n! m=0 m! vacuum” in quantum-field theory parlance) from the state. We m =n call such states “vacuum-removed states” for short. The sufficient condition (8) then implies the following condi- In summary, our results indicate that any δ-punctured ther- tion on w1: mal state with w satisfying Eq. (16) corresponds to a proper 1 2 n physical state. As can be seen in Fig. 1, the value of the bound |α1| n¯ √ e 1 n¯+1 δ,SC | | w1 < min n! ≡ w , (22) of allowed weights w1 decreases as α1 increases, showing + | | ∈ | | max n¯ 1 f ( α1 ) n N α1 that a thermal state cannot be punctured significantly far from its center. This feature is less pronounced as n¯ increases. where +∞ m |α1| f (|α1|) = √ . (23) B. Conditions for nonclassicality m! m=0 Negativity of the P function. All δ-punctured thermal states δ,SC are nonclassical due to the presence of the infinite negative δ The bound wmax depends only on the absolute value of α1, δ,SC = = peak in the P function. not on its phase. Also, we have wmax 0 only if n¯ 0. The Negativity of the Wigner function. The Wigner function largest values of w1 allowed by the bound lie in the region of δ-punctured thermal states follows from Eq. (9) and is where |α1| and n¯ are small, which is shown in Fig. 1. Since δ,SC given by wmax > 0 for all values of n>¯ 0 and α1, this is an analytical proof that there always exist physical nonclassical δ-punctured 2|α|2 − + thermal states. 2N e 1 2n¯ 2 −2|α−α1| δ,NC W(α) = − w1 e . (24) Tightness of wmax . Computations based on the diagonal- π 1 + 2n¯ ization of δ-punctured thermal states and the application of NSC δ,NC suggest that the bound wmax given in Eq. (16) is tight. It takes negative values for large enough puncture weight This claim is further supported by the observation that the + | n¯ 1 −|α |2/n¯ coherent state ( n¯ )α1 from which the bound is derived is e 1 = δ,NC w1 > . (25) an eigenstate of ρ with w1 wmax with eigenvalue zero. 1 + 2n¯
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where S(r) = exp (−r[a†2 − a2]/2) is the squeezing operator and r is the squeezing parameter, taken real for simplicity. We recall that the P function of a state ρ is given by the Fourier transform of its normal-ordered characteristic function χ(β) = i(βa†+β∗a) tr[ρe ]. For the squeezed thermal state ρcl defined in Eq. (27), the normal-ordered characteristic function reads [38]
−n¯ β2 −n¯ β2 χ(β) = e I R R I , (28)
where βR and βI are the real and imaginary parts of β ∈ C and −2r −r n¯ R = e n¯ − e sinh r, 2r r n¯ I = e n¯ + e sinh r. (29) The Fourier transform of the characteristic function—and thus the P function—exists only if both the coefficients n¯ I and n¯ R 2 2 in front of βR and βI are positive, which imposes the condition e−2|r| n¯ − e−|r| sinh |r| > 0, (30) thereby limiting the range of allowed squeezing parameters r ∈ R for a given n¯. The characteristic function of a squeezed thermal state is an asymmetric Gaussian distribution; the same goes for its P function. Therefore, a δ-punctured squeezed thermal state is a state determined through (1) and (2)with 2 2 − αR + αI e n¯R n¯I Pcl(α) = √ , π n¯ Rn¯ I (31) π(α) = δ(α), FIG. 2. Density plot of the second-order correlation function g(2) | | [Eq. (26)] for single δ-punctured thermal states as a function of α1 where αR and αI are the real and imaginary parts of α ∈ C. = δ,NC and n¯ for the maximal puncture weight w1 wmax given by Eq. (16) (top) and as a function of |α | and w for n¯ = 0.2 (bottom). Only 1 1 A. Conditions for (semi)positive definiteness the range where ρ 0 is shown. The black dashed curve in the top (2) figure, along which g = 1, delimits the sets of states displaying Necessary condition. Since Pcl(α) is rotationally nonsym- antibunching or not. The inset of the bottom figure represents g(2) as metric, it is reasonable to assume that the necessary condi- = δ,NC a function of α1 for the maximum weight w1 wmax . tion (7) will provide us with a tight bound for w1 only if we consider states |φ ∈K with nonsymmetric P functions. By K Such weights are still acceptable as long as they do not exceed taking to be the set of squeezed coherent states [38,39], we the (larger) bound (16). obtain the following upper bound for the puncture weight w1: Antibunching. For δ-punctured thermal states, the second- α2 α2 √ − 1R + 1I order correlation function (10) is given by 2 n¯ n¯ e n¯R n¯I w R I ≡ wδ,NC . 1 + + max, squ (32) 2n¯ 2 − w |α |4 n¯ R n¯ I 2n¯ Rn¯ I g(2) = (1 − w ) 1 1 . (26) 1 2 2 (n¯ − w1|α1| ) When n¯ R = n¯ I = n¯, this condition reduces to the condi- (2) tion (16), as expected. For w → 0, we recover the known value g = 2ofthe 1 Proof. Let us denote by |γ,r a squeezed coherent state thermal state [37]. In Fig. 2, we show a density plot of g(2) defined as [38,39] as a function of n¯ and |α1| for the maximal allowed value of the puncture weight w1, i.e., the bound (16). A whole region |γ,r =D(γ )S(r )|0 , (33) of parameter space corresponds to punctured states giving rise † ∗ to antibunching (g(2) < 1). Also, g(2) increases beyond 2 for where D(γ ) = exp (γa − γ a) is the displacement operator n¯ 0.1 and |α |0.5. and S(r ) the squeezing operator of parameter r , where the 1 prime symbol distinguishes r from r, this latter defining n¯ R | =| δ and n¯ I . The necessary condition (7) with φ γ,r reads IV. -PUNCTURED SQUEEZED THERMAL STATES P (α)|γ,r|α |2d2α To generalize the previous results, we now replace the w cl , (34) 1 2 thermal state by a squeezed thermal state [38,39] |γ,r |α1 | 1 −|α|2/n¯ † 2 where the modulus squared of the scalar product between a ρ = e S(r)|α α|S (r) d α, (27) cl πn¯ squeezed coherent state |γ,r and a coherent state |α is given
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FIG. 4. Density plot of the second-order correlation function g(2) [Eq. (39)] for single δ-punctured squeezed thermal states as a function of n¯ R and n¯ I for the maximal√ puncture weight w1 given in Eq. (32) = = wδ,NC for α1R α1I 0.1/ 2. The black dashed curve delimits the region FIG. 3. Plot of the upper bound max, squ (solid curves) and of (2) (2) δ where g < 1. Note that in the absence of puncture, g > 2forany w ,thew1 which cancels the smallest eigenvalue of the density max, squ squeezed thermal state (see text). matrix, as a function of |α1| for n¯ R = 1, n¯ I = 0.1 (squares), and n¯ I = 0.5 (dots). The eigenvalues are computed in a Fock space of = δ dimension nmax 25. Inset: convergence of wmax, squ (green squares) and takes negative values if the puncture weight satisfies with increasing cutoff dimension nmax (nmax = 6, 8, 25 from top to α2 α2 bottom) for n¯ I = 0.1. − R + I e n¯R n¯I w > , (38) 1 1 + 2n¯ by [37] which is still acceptable as long as it does not exceed the (larger) bound (32). 2 −(|α|2+|γ |2)+(α∗γ +γ ∗α)sech(r) |γ,r |α | = sech(r ) e Antibunching. For δ-punctured squeezed thermal states, the × −th(r)Re[α2−γ 2] second-order correlation function (10) is given by e . (35) 1 3n¯ 2 + 2n¯ n¯ + 3n¯ 2 − w |α |4 (2) = − 4 R R I I 1 1 The minimum of the RHS term in Eq. (34) is obtained, after g (1 w1) + 2 . (39) n¯ R n¯ I − | |2 integration over α,for 2 w1 α1 For w → 0, we recover the known value for squeezed thermal + + 1 n¯ R n¯ I 2n¯ Rn¯ I states [38] γ = √ α1, (n¯ R − n¯ I + 2n¯ Rn¯ I )(n¯ I − n¯ R + 2n¯ Rn¯ I ) 3n¯ 2 + 2n¯ n¯ + 3n¯ 2 − g(2) = R R I I n¯ R n¯ I 2 r = arctanh , (36) (n¯ R + n¯ I ) 2n¯ Rn¯ I (2n¯ + 1)2 sinh2 r cosh2 r = 2 + 2. (40) and leads to the necessary condition (32). (n¯ cosh(2r) + sinh2 r)2 δ,NC Tightness of wmax, squ. Computations relying on the diago- (2) nalization of ρ given by Eq. (1) with Eqs. (2) and (31) suggest Figure 4 shows a density plot of g as a function of n¯ R δ,NC and n¯ I for |α1|=0.1 with the maximal allowed value of that the bound wmax, squ giveninEq.(32) is tight. As for the case of thermal states, this claim is supported by the observation w1, i.e., the bound (32). Again, a whole region of parameter | space (delimited by the black dashed curve) corresponds to that the squeezed coherent state γ,r with γ and r given by (2) Eq. (36) is an eigenstate of ρ with eigenvalue zero. Figure 3 states displaying antibunching (g < 1), to be contrasted with shows wδ,NC and the value of w that cancels the smallest squeezed thermal states (i.e., without puncture) for which max, squ 1 (2) eigenvalue of ρ. Numerical results converge to the bound as g 2. the cutoff dimension increases. V. GAUSSIAN-PUNCTURED THERMAL STATES
B. Conditions for nonclassicality We now generalize the results of Sec. III by replacing δ punctures by single narrow Gaussian punctures. We thus Negativity of the P function. As for δ-punctured thermal define Gaussian-punctured thermal states as states determined states, the presence of a δ peak always enforces nonclassicality. through (1) and (2) with Negativity of the Wigner function. The Wigner function of a δ-punctured squeezed thermal state reads e−|α|2/n¯ P (α) = , cl πn¯ 2 2 − αR + αI (41) 2 1+2n 1+2n −| |2 2N e R I − | − |2 α /b = − 2 α α1 e W(α) w1 e , (37) π(α) = , π 1 + 2n¯ πb
023832-6 NONCLASSICAL STATES OF LIGHT WITH A SMOOTH … PHYSICAL REVIEW A 97, 023832 (2018) where b>0 characterizes the width of the puncture. Note that Negativity of the Wigner function. The Wigner function of δ-punctured thermal states correspond to the limit b → 0. a vacuum-centered Gaussian-punctured thermal state reads ⎛ ⎞ − 2|α|2 − 2|α|2 A. Vacuum-centered Gaussian punctures 2 ⎝ e 1+2n¯ e 1+2b ⎠ W(α) = N − w1 (46) We first consider the simple case of a thermal state from π 1 + 2n¯ 1 + 2b which we subtract a single Gaussian centered at α1 = 0. A thermal state with the vacuum component exactly removed and takes negative values if w1 satisfies (case b → 0, n¯ = 1,w1 = 1/2) has already been considered 1 + 2b in the literature as a simple example of a nonclassical state [40]. w > , (47) Here we keep a variable width and amplitude for the subtracted 1 1 + 2n¯ Gaussian. which is a stronger condition than Eq. (45). Note that for b → 1. Condition for (semi)positive definiteness 0, Eq. (47) tends to Eq. (25), the bound found for δ-punctured thermal states. Necessary and sufficient condition. Since both the original Antibunching. From the expression of the second-order and the subtracted state are diagonal in the Fock state basis, correlation function (10), we have NSC can be tackled analytically. The P function of the vacuum-centered Gaussian-punctured thermal state reads 2n¯ 2 − w 2b2 (2) = − 1 (48) g (1 w1) 2 . − |α|2 − |α|2 (n¯ − w1b) e n¯ e b P (α) = N − w . (42) 1 = G, NSC πn¯ πb For w1 wmax, vac, the condition for antibunching reads = It takes negative values at α 0 as soon as w1 >b/n¯.Aswe g(2) < 1 ⇔ n¯ 2 + b2 < 1. (49) will show, only b
→ + | |2 showing that, in the limit b 0, we recover the fact that all b 1 − α1 G, NC w e n¯−b ≡ w . (52) δ-punctured states are nonclassical. 1 n¯ + 1 max
023832-7 DAMANET, KÜBLER, MARTIN, AND BRAUN PHYSICAL REVIEW A 97, 023832 (2018)
Proof.Using|γ |α |2 = e−|α−γ |2 and Eq. (41), we find for 1 the expectation value of ρ in the coherent state |γ 1 0.8 | |2 | − |2 0.8 1 − α −|α−γ |2 w1 − α α1 −|α−γ |2 2 0.6 ργγ = e n¯ − e b d α. G πn¯ πb max 0.4 w (53) 0.6 0.2 Evaluation of the Gaussian integrals leads to 0 G max 0 0.5 1 1.5 2 2 | − |2 b + 1 − |γ | + γ α1 w ρ 0 ⇔ w e n¯+1 b+1 . (54) 0.4 |α1| γγ 1 n¯ + 1 That needs to be true for all γ ∈ C in order to keep the possi- 0.2 bility of a positive semidefinite density operator. Minimizing G, NC this expression over γ gives us the upper bound wmax . Since the exponential function is strictly monotonically increasing 0 and the prefactors are positive we can focus on minimizing the 0 0.5 1 1.5 2 |α | exponent: 1 2 2 |γ | |γ − α1| G, NC G f γ ≡− + . FIG. 5. Plot of the upper bound wmax (solid curve) and of wmax, ( ) + + n¯ 1 b 1 the w1 which cancels the smallest eigenvalue of the density matrix, as a | | = = = By applying a rotation in phase space, the puncture’s center function of α1 for b 1andn¯ 1.5 (squares) and n¯ 2 (dots). The = can always be brought along the real axis, so that we can set eigenvalues are computed in a Fock space of dimension nmax 40. Inset: convergence of wG (green squares) as a function of the cutoff α1 = x1 ∈ R. We then split γ into its real and imaginary part max dimension nmax (nmax = 8,12,40 from top to bottom). γ = γR + iγI , leading to x2 f (γ ) = h(γ ) + g(γ ) + 1 , the numerical bound (squares and dots) for different cutoff I R b + 1 dimensions nmax as a function of the center of the puncture with |α |. Also, we found that the coherent state |γ with γ = 1 (n¯ + 1)/(n¯ − b) α1 is an eigenstate of ρ with eigenvalue zero 2 1 1 h(γI ) = γ − , when w is given by the bound (52). I b + 1 n¯ + 1 1 1 1 2γ x g γ = γ 2 − − R 1 . 2. Conditions for nonclassicality ( R) R + + + b 1 n¯ 1 b 1 Negativity of the P function. From Eq. (51), we see that Here we have to distinguish two cases (the case b = n¯ is partly negative P functions can be obtained when trivial). (i) If b>n¯, h(γ ) is an inverted parabola and can attain I | |2 b α1 − − arbitrary negative values and the same holds for f , thus pushing w1 > e n¯ b , (55) the exponential in (54) arbitrarily close to zero, implying that n¯ the above condition can only be fulfilled for all coherent states which generalizes Eq. (45). Hence the lower bound of the if w1 = 0. This means that we cannot subtract any Gaussian weight decreases with the distance between the center of the wider than the original thermal one and still obtain a valid state. thermal state and the position of the puncture. Further this is G, NC This generalizes what we have already seen in the case of the smaller than wmax given in (52) since n>b¯ , implying that vacuum centered Gaussian. (ii) For b
023832-8 NONCLASSICAL STATES OF LIGHT WITH A SMOOTH … PHYSICAL REVIEW A 97, 023832 (2018)
q =1 q =2 q = q |0 , |2 , ... |0 , |4 , ... |0 , |2q ,...
ρT
|1 , |3 , ... |2 , |6 , ... |q , |3q ,...
ρ
FIG. 7. Sketch of realization of vacuum-removed thermal state. A thermal state ρT is subjected to multiple QND measurements that distinguish the vacuum state of other photon number states. If at least one of the measurements yields “−”, the vacuum has definitely not been realized and the measured state is added to the ensemble. If FIG. 6. Density plot of the second-order correlation function g(2) all measurements yielded “+” the state is discarded. Since we only as a function of n¯ and b 023832-9 DAMANET, KÜBLER, MARTIN, AND BRAUN PHYSICAL REVIEW A 97, 023832 (2018) state is diagonal in the Fock basis; all coherences between we therefore present another, completely different approach, different Fock states are lost. In such a case, however, the based on approximately synthesizing our states from scratch procedure is very effective: if we want to make sure that states as follows: we can numerically express the density operator with photon number less than n0 were not truncated, we only of the desired punctured state up to a chosen dimension N in = need to perform lmax log2(n0) different measurements. the Fock basis, neglecting terms involving states with higher For a thermal state, occupation for high photon number photon number than N: states decays very rapidly for small temperatures, and the N N overall error can be kept small. To quantify this, we can ρ = c |n m|. calculate the fidelity of our desired state (59) with respect to mn n=0 m=0 the state that was actually constructed This is in general a mixed state and we can express it as a sum ∞ n¯ 2lmax n of pure states: lmax lmax ρ = N ρT − |2 n 2 n| . (60) (1 + n¯)2lmax n+1 n=0 ρ = pi |ψi ψi |, = Since both density operators are diagonal in Fock basis the i 1 fidelity is simply evaluated as [46, p. 409] N |ψ = c(i)|n . ∞ i n = = − 1 1 n=0 F (ρ,ρ ) ρnnρnn 1 l . (61) n¯+1 2 max = n¯ − We can then create our state by choosing with the right proba- n 0 n¯ 1 bility one of the pure states and constructing it with the method In the limit of large lmax this reduces to proposed in [49–52] and demonstrated experimentally in [53]. 2lmax − log( n¯+1 )2lmax Multiple repetition (creation of an adequate ensemble of states) 1 n¯ e n¯ F = − = − , then gives us our state up to a certain accuracy. In a single run 1 + 1 (62) n¯ n¯ 1 n¯ we actually do not have the complete state, but for further experiments one anyway needs typically multiple repetitions i.e., the fidelity approaches 1 exponentially with 2lmax . This shows the theoretical viability of the procedure. Exper- for obtaining measurement statistics of any observable. Hence imental imperfections, as for example the nonperfect linearity this procedure creates the desired mixed state in the ensemble of the phase shift in the photon numbers [43], may worsen the sense. result and will have to be evaluated for a given experimental setup. VII. CONCLUSION In this work, we introduced a class of nonclassical states B. “Vacuum or not” measurements with regular nonpositive Glauber-Sudarshan P function that The vacuum state can be removed from a broader class of we call punctured states. These states are obtained from states, namely states that do not contain coherences between the addition of sufficiently narrow negative peaks to smooth positive P functions. We determined the regimes of parameters the vacuum state and other Fock states. Indeed, if ρnn = 0in yielding proper physical (i.e., positive semidefinite) nonclas- the Fock basis, then ρ0n = ρn0 = 0 is implied by the positivity of the state [47]. Then one can remove the vacuum state with sical states in the case of δ and Gaussian punctures of thermal a measurement operator A of the form or squeezed thermal states. We showed that their second-order correlation function can be modified through an appropriate ∞ choice of the punctures and identify the regimes yielding = | |+ | | A a0 0 0 a1 n n , (63) antibunching of light. All states exhibiting antibunching have n=1 anegativeP function in some region. Finally, we presented with a0 = a1: when a0 is measured one discards the state, for a1 possible experimental realizations of punctured states based on one continues the experiment and effectively gets the desired vacuum-or-not measurements and on complete synthesizing. punctured state through postselection. Recently, a procedure was proposed that realizes the measurement operator A [48]. ACKNOWLEDGMENTS F.D. acknowledges fruitful discussions with Daniel K. L. C. Synthesizing arbitrary states Oi. 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