aa r X i v : . [ qu a n t - ph ] F e b Positive energy density leads to no squeezing
S. Kannan ∗ and C. Sudheesh † Department of Physics, Indian Institute of Space Science and Technology, Thiruvananthapuram, 695 547, India. (Dated: February 9, 2021)We consider two kinds of superpositions of squeezed states of light. In the case of superpositionsof first kind, the squeezing and all higher order squeezing vanishes. However, in the case of thesecond kind, it is possible to achieve a maximum amount of squeezing by adjusting the parametersin the superposition. The emergence and vanishing of squeezing for the superposition states areexplained on the basis of expectation values of the energy density. We show that expectation valuesof energy density of quantum states which show no squeezing will be always positive and that ofsqueezed states will be negative for some values of spacetime-dependent phase.
I. INTRODUCTION
Squeezed states are a class of nonclassical states of lightwhich have less fluctuations in one quadrature phase thana coherent state at the expense of increased fluctuationsin the other [1–4]. These states offer intriguing appli-cations such as in optical communication systems [5, 6],quantum sensing [7], noise-free amplification [8], gravita-tional wave detection [9–11] etc. Recently the increasein sensitivity using squeezed light in LIGO and VIRGOdetectors has been reported [12, 13]. A rapid increasein the use of squeezed states in the continuous variablequantum information processing is also seen [14]. Engi-neering interactions between electric dipoles utilizing theantisqueezed quadrature of a squeezed vacuum state hasbeen predicted recently. This does not require any pho-tonic structures [15]. As many such theoretical and prac-tical implications exist, investigations of squeezing prop-erties of various quantum states is still a central topic inquantum optics.As pointed out by Dirac, the linear superposition prin-ciple is one of the most fundamental features of quantummechanics. It has been shown that quantum superpo-sition can give rise to various nonclassical effects suchas squeezing, higher-order squeezing [16], sub-Poissonianstatistics, etc. Superposition of two coherent states [17],superposition leading to even and odd squeezed coherentstates [18, 19], superposition of squeezed coherent states[20], binomial state [21, 22], intermediate number-phasestate [23] and generalized superposition of two squeezedstates [24] are some examples for the nonclassical states.Despite the fact that there are a lot of studies in theliterature which talk about squeezing due to superposi-tion of states but there is no work which clearly stateswhat type of superpositions leads to squeezing and itsconnection with any other physical parameter of the sys-tem. However, there are some attempts has been madein the literature [17, 25] to address the aforementionedquestions. In this paper, we analyze different types ofsuperposition states to find what type of superpositions ∗ Present address: ISRO Inertial Systems Unit, Thiruvananthapu-ram, 695 013, India.; [email protected] † [email protected] give rise to squeezing, and to find a relation betweensqueezing and expectation value of the energy density.Another motivation to do this study is to devise a methodto get higher or no squeezing when we add two squeezedstates. These two analyses will definitely help researchersworking in quantum communication and quantum sens-ing using squeezed light to choose proper squeezed statesfor their study. For this purpose, we consider the super-position of various squeezed states such as the squeezedvacuum states [3], photon-added coherent states [26] etc.We investigate both the nonclassical and statistical fea-tures of the states and identify the role of superposition inmodifying the quantum fluctuations in the field quadra-tures.The paper is organized as follows: in Section II westudy the squeezing and higher-order squeezing of su-perposition of various squeezed states, vanishing and en-hancing of squeezing for different superposition is seen.In Section III expectation value of energy density is usedto explain the squeezing properties shown by the super-position states and Section IV is devoted to conclusion. II. SUPERPOSITION OF SQUEEZED STATESAND ITS SQUEEZING
Squeezing can be quantified by analyzing the uncer-tainties in the quadrature operators. For a single-modescenario, the quadrature operators are defined asˆ X = (ˆ a + ˆ a † ) / , ˆ P = (ˆ a − ˆ a † ) / i. (1)They satisfy the commutation relation[ ˆ X, ˆ P ] = i/ h (∆ ˆ X ) ih (∆ ˆ P ) i ≥ / . (3)(We have set ~ = 1.) A given state is said to be quadra-ture squeezed if the variance in one of the quadraturefalls below the vacuum value: h (∆ ˆ X ) i < / h (∆ ˆ P ) i < / . (4)Here we only consider the superposition of the same classof quadrature squeezed states. We can classify the super-position of these types of states into two: (1) all the statesin the superposition have the same squeezing value, (2) ageneralized superposition where the states have differentsqueezing values. The first kind of superposition can beexpressed as | Ψ i l = N l l − X j =0 (cid:12)(cid:12)(cid:12) Φ e i πj/l E , (5)where N l is the normalization constant and | Φ i can beany squeezed state of the same class. Here, it is alwayspossible to find an observable which can turn one of thestates in the superposition to another one in the super-position. The second kind, the generalized superpositionis of the form: | Ψ i gen = N gen l − X j =0 a j | Φ j i . (6)Here N gen denotes the normalization factor, | Φ j i repre-sents squeezed states of the same class, and a j the corre-sponding weight factors.Let us now analyze the quadrature variance of super-positions of some important squeezed states. A. Superpositions of the first kind
1. Squeezed vacuum state
The squeezed vacuum state | ξ i is generated by the ac-tion of squeezing operator ˆ S ( ξ ) [2–4] on the vacuum state: | ξ i = ˆ S ( ξ ) | i , (7)where ξ = re iθ with real r (squeezing parameter) and0 ≤ θ ≤ π . Experimental realization of this state wasachieved way back in 1985-1986 [27, 28]. In the recentyears, great progress in the generation of this state havebeen achieved [29–32] that could significantly improvequantum information applications.In the Fock basis, | ξ i can be expanded as | ξ i = 1 p cosh(r) ∞ X m =0 (cid:0) − e iθ tanh(r) (cid:1) m p (2 m )!2 m m ! | m i . (8)For this state, the variance in the both quadratures comeout to be [33] h (∆ ˆ X ) i = 14 e − r and h (∆ ˆ P ) i = 14 e r , (9)and, hence, there is a squeezing in the ˆ X -quadrature.Now, consider a superposition of l squeezed vacuum state | ζ i l of the first kind: | ζ i l = N s l − X j =0 (cid:12)(cid:12)(cid:12) ξe i πj/l E , (10) where N l is the normalization constant and is equal to N l = ∞ X m =0 (2 lm )!2 lm ( lm )! (tanh(r)) lm ! − / . (11)In the Fock basis, | ζ i l = N l ∞ X m =0 (cid:0) − e iθ tanh(r) (cid:1) lm p (2 lm )!2 lm ( lm )! | lm i . (12)The above superposed states can be produced by utilizingthe techniques in [34] . We find that for the states | ζ i l ≥ , h ˆ a i = h ˆ a † i = 0 (13)and h ˆ a i = h ˆ a † i = 0 . (14)Since the above above expectation values are zero, we get h (∆ ˆ X ) i = h (∆ ˆ P ) i = 14 + N l ∞ X m =0 (tanh(r)) lm lm (2 lm )!2 lm ( lm )! . (15)Since the RHS of the above equation is always greaterthan 1 /
4, there is no squeezing for | ζ i l ≥ in both thequadratures. It is one of the important results of this pa-per. To check whether this is true for any superpositionof squeezed states of first kind, we examine superposi-tions of other types of squeezed states in the followingsubsections.
2. Photon-added coherent state m -photon-added coherent states (m-PACS) were intro-duced as the states which are intermediate between theFock states and the coherent states. The Fock state rep-resentation of m -PACS is | α, m i = e −| α | / q L m ( − | α | ) m ! ∞ X n =0 α n n ! p ( n + m )! | n + m i , (16)where L m is the Laguerre polynomial of order m , α (co-herent parameter) is a complex number and m is an in-teger. These states show rich nonclassical properties andexhibit quadrature squeezing for a wide range of α values[26]. Experimental generation of the single photon-addedcoherent state has been achieved [35], and there also ex-ists schemes for the generation of higher-order photon-added coherent states [36, 37]. Consider the l superposi-tion of the first kind of m -PACS: | Γ , m i l = N m l − X j =0 (cid:12)(cid:12)(cid:12) αe i πj/l , m E , (17)where N m is the normalization constant. Since we areexamining only superposition of squeezed states, we con-sider range of α for which the state | α, m i shows quadra-ture squeezing. We then calculate the quadrature vari-ance for the state | Γ , m i l for different m and l values.Our calculations show that the quadrature squeezing isalso absent for all m values for the state | Γ , m i l ≥ . Weconfirm our result also for superposition of two-modesqueezed states.
3. Two-mode squeezed vacuum state
Quadrature squeezing is observed in multi-mode sce-narios also. The most common one is the two-modesqueezed vacuum state | ξ i . It is generated by the ac-tion of a two-mode squeeze operator ˆ S ( ξ ) [38] on thevacuum state: | ξ i = ˆ S ( ξ ) | , i = 1cosh(r) ∞ X n =0 (cid:0) − e iθ tanh(r) (cid:1) n | n, n i , (18)where ξ = re iθ . Since there is a correlation betweenthe modes, squeezing exist in the effective quadratureoperators:ˆ X = (ˆ a + ˆ a † + ˆ b + ˆ b † ) / / , ˆ X = (ˆ a − ˆ a † + ˆ b − ˆ b † ) /i / . (19)Here also we consider the first kind of superposition andvariance in the above quadratures operators are analyzed.For the two superposition state N ( | ξ i + |− ξ i ) = 2 N cosh(r) ∞ X n =0 e inθ (tanh(r)) n | n, n i , (20)with normalization constant N , the variance in thequadratures ˆ X and ˆ X comes out to be the same and isequal to∆ ˆ X = ∆ ˆ X = 14 (cid:18) N cosh(r) (cid:19) ∞ X n =0 n (tanh(r)) n ! . (21)So there is no squeezing in the superposition state andwe confirm also the absence of quadrature squeezing forall higher numbers (i.e., l >
2) of superpositions. Inthe above three cases, we have shown that no first-ordersqueezing exists when we take superposition of the firstkind with squeezed states. In the next section, we extendour analysis for higher-order squeezing of superpositionof squeezed states.We have also calculated both the Hong and Mandel[16] and Hillery type [39] higher order squeezing for thesuperposition of squeezing states. Our calculations showsthat there is no higher-order squeezing exists for the su-perposition of the states which we have considered ear-lier. In this section, we have shown that superposition ofsqueezed states can lead to complete vanishing of squeez-ing properties. This is quite unexpected, since usuallythe superposition of wavepackets give rise to squeezingproperties (e.g. even-cat state [40]).
B. Generalized superposition
We have defined the generalized superposition state | Ψ i gen in Eq. 6 as the superposition of squeezed stateswith different squeezing value and weight factors. Here,we analyze the variance and higher-order moments in thequadrature of | Ψ i gen for squeezed vacuum states: | Ψ i gen = N gen l − X j =0 a j | ξ j i , (22)where ξ j = r j e iθ j , a j ’s are the weight factors and thenormalization constant N gen = l X i =1 l X j =1 a i a ∗ j p cosh( r i − r j ) − / . (23)The squeezing and statistical properties of the two su-perposition cases have been discussed by Barbosa et al.in [24]. Here, we also find squeezing properties of morethan two superpositions and in turn find suitable weightfactors to achieve maximum quadrature squeezing. Thevariance in the ˆ X quadrature for the state given in Eq.22 comes out to be h (∆ ˆ X ) i = 14 + N gen " l X i =1 l X j =1 ∞ X n =0 a i a j (tanh( r i )tanh( r j )) n p cosh( r i )cosh( r j ) × (2 n )!2 n ( n !) n − (2 n + 1) (cid:16) tanh( r i ) + tanh( r j ) (cid:17)! . (24)Without loss of generality, we take real values for ξ j ’s and a j ’s. Using a minimization algorithm [41], the weight fac-tors ( a i ’s) are calculated to achieve the highest squeez-ing. For fixed squeezing parameters, the weight factorsare computed such that the variance attains the min-imum possible value. We found that the state | Ψ i gen can have quadrature variance less than 14 e − R , where R = max( r j ). The minimum value of variance obtainedfor different numbers of superpositions along with thecorresponding weight factor is tabulated below.It is clear from the table that it is possible to increasethe squeezing by increasing the superposition in the state.We have also studied the higher-order squeezing the stateand found that the state exhibits higher-order squeezing.For example, the state | Ψ i gen with l = 3, the fourth andsixth-order squeezing comes out to be h (∆ X ) i = 0 . h (∆ X ) i = 0 . TABLE I.S/N Squeezing parameters Variance Weight factors( r j ) h (∆ ˆ X ) i ( a j )1 l = 1, r = 1 0.0338 a = 12 l = 2, r = 0 . r = 1 0.0268 a = − . a = 13 l = 3, r = 0 . r = 0 . r = 1 0.0188 a = − . a = − . a = 14 l = 4, r = 0 . r = 0 . r = 0 . r = 1 0.0151 a = − . a = 2 . a = − . a = 1 leads us to find a connection between energy density andsqueezing which we describe in detail in the next section. III. SQUEEZING AND ENERGY DENSITY
An obvious question that arises now is, why is thesqueezing in some superposition state vanishing? Or whyis it increasing in certain cases? It is well known thatsome superposition of coherent state, such as even catstate [40] and Yurke-Stoler states [42] exhibit quadra-ture squeezing while coherent state doesn’t have such anonclassical property. In this section, we explain theseproperties based on the expectation value of the energydensity of these quantum states.It is known that all forms of classical matter have non-negative energy density but this is not the case for a gen-eral quantum state. A superposition of number eigen-states can give a negative expectation value of energydensity in certain spacetime regions due to coherence ef-fect. To find the expectation value of the energy density,we calculate the expectation value of the stress-energytensor or energy-momentum tensor [43]. Stress-energytensor describes the density and flux of energy and mo-mentum in spacetime. In order to calculate this expec-tation value for the stress-energy tensor, the method de-scribed in reference [44] is followed. Consider a masslessand minimally coupled scalar field. Its Lagrangian den-sity is given by L = 12 η µν ( ∂ µ φ )( ∂ ν φ ) , (25)where the spacelike metric η µν = diag[-1,1,1,1] and φ isthe quantum field operator which satisfies the dynamicalequation (cid:18) − ∂ ∂ t + ∇ (cid:19) φ ( x ) ≡ ∂ µ ∂ µ φ ( x ) = 0 . (26)We have taken c = ℏ = 1.The quantum field operator φ can also be expanded in mode functions as φ = X k ( a k f k + a † k f ∗ k ) , (27) where a k satisfy the usual bosonic commutation relationand the mode function f k is given by f k = (2 L ω ) − / e i k.x − iωt . (28)Here we have assumed periodic boundary condition in athree-dimensional box of side L and ω is the frequencyof the mode and k the wave number. For our purpose,we consider the stress tensor corresponding to the singlemode excitation: T µν = ( ∂ µ φ )( ∂ ν φ ) − η µν ( ∂ σ φ )( ∂ σ φ ) . (29)We calculate the normal ordered (same as subtractingthe vacuum expectation value) expectation value of thisstress tensor for the quantum states of our interest. Wewill be only looking at the expectation value of the tem-poral component ( h : T : i ). For coherent state | α i , thisnormal ordered expectation value comes out to be h : T : i = 2 K α (1 − cos(2 θ )) , (30)where K = ( k ) ωL and the spacetime-dependent phase θ = k ρ x ρ . Without loss of generality, the above expres-sion is derived for real values of α . It is easy to see thatfor coherent states, h : T : i ≥ α value for all values of θ . For even cat state | ψ i = 2 q e − | α | ) ∞ X n =0 α n p (2 n )! | n i , (31)with real α , the energy density comes out to be h : T : i = 2 K α tanh( α ) (cid:16) − coth( α )cos(2 θ ) (cid:17) . (32)For fixed values of α , the energy density becomes neg-ative for some values of θ in [0 , π ]. Thus for the evencat state, which shows quadrature squeezing, energy den-sity becomes negative for certain values of spacetime-dependent phase θ . In contrast, we have found earlierthat a coherent state, which shows no squeezing, nevergives negative energy density for any values of θ . Tocheck that whether the above result is a general resultapplicable to all quantum states, we carry out the aboveanalysis to other important class of states and superpo-sition states considered in the Section II.The energy density for odd cat state, which shows nosqueezing, is always positive. For the squeezed vacuumstate defined in Eq. 8the average energy density h : T : i = 2 K sinh(r) (cid:16) cosh(r)cos(2 θ ) + sinh(r) (cid:17) . (33)Again, we have considered real values for ξ (= r ) withoutloss of generality. Here, the energy density can go belowzero for certain values of θ for a given r and for largervalues of r the energy density tends to zero. This can beassociated with larger oscillations in the photon numberdistribution [45]. For the superposition state of first kindgiven in Eq. 10 with l = 2, which shows no squeezing,the average energy density h : T : i = 16 K sech(r) (cid:16) p sech(2r) (cid:17) (cid:16) ∞ X n =0 n (4 n )!(2 n )! (tanh(r) / n (cid:17) (34)is always greater than zero. Higher order superpositionwill also give rise to a positive average energy density.Our results are also verified for PACSs (Eq. 16) andtheir superpositions. Finally for the generalized statesgiven in TABLE. I, the expectation value for the stressenergy tensor is found to have negative values.One of the important results from this section can bestated as follows: The expectation value of energy den-sity of quantum states which show no squeezing will bealways positive and that of squeezed states will have neg-ative for some values of spacetime-dependent phase. Ouranalysis also gives the reason for absence of squeezing insuperposition states of first kind and presence of squeez-ing in superposition of states of second kind in terms ofexpectation value of the energy density. IV. CONCLUSION
We have studied the squeezing properties of arbitrarynumbers of superpositions of various squeezed states suchas squeezed vacuum state, photon-added coherent statesand two mode squeezed vacuum state. We have con-sidered two kinds of superpositions: first kind and sec-ond kind. In the case of first kind, the superpositions ofsqueezed states doesn’t show both squeezing and higher-order squeezing of all orders. This is found to be truefor any state which has quadrature squeezing and multi-mode squeezed states. However, in the case of the super-positions of second kind (also called as generalized su-perpositions), it has been shown that the superpositionstates show large amounts of quadrature and higher-ordersqueezing. This is achieved by choosing the proper weightfactors in the superpositions; this method also enables usto have a control over the amount of squeezing produced.The vanishing and appearance of squeezing in super-position state is explained on the basis of expectationvalues of energy density or stress-energy tensor. Stateswith squeezing are shown to have a negative expecta-tion value for the stress-energy tensor for some values ofspacetime-dependent phase. In the case of states withno squeezing, the expectation values of energy densityis always positive. We have found that in the case ofsuperposition of squeezed states of first kind, the expec-tation values of energy density is always positive and inthe case of second kind, the average value is negative forsome values of spacetime-dependent phase. [1] B. R. Mollow and R. J. Glauber, Physical Review ,1076 (1967).[2] D. Stoler, Physical Review D , 3217 (1970); , 1925(1971).[3] H. P. Yuen, Physical Review A , 2226 (1976).[4] D. F. Walls, Nature , 141 (1983).[5] H. Yuen and J. Shapiro, IEEE Transactions on Informa-tion Theory , 657 (1978).[6] J. Shapiro, H. Yuen, and A. Mata, IEEE Transactionson Information Theory , 179 (1979).[7] B. J. Lawrie, P. D. Lett, A. M. Marino, and R. C. Pooser,ACS Photonics , 1307 (2019).[8] M. Majeed and M. S. Zubairy, Physical Review A ,4688 (1991).[9] C. M. Caves, Physical Review D , 1693 (1981).[10] J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott,M. Abernathy, C. Adams, R. Adhikari, C. Affeldt,B. Allen, G. S. Allen, et al. , Nature Physics , 962 (2011).[11] J. Aasi, J. Abadie, B. P. Abbott, R. Abbott, T. D. Ab-bott, M. R. Abernathy, C. Adams, T. Adams, P. Ad-desso, R. X. Adhikari, et al. , Nature Photonics , 613(2013).[12] M. Tse, H. Yu, N. Kijbunchoo, A. Fernandez-Galiana,P. Dupej, L. Barsotti, C. D. Blair, D. D. Brown, S. E.Dwyer, A. Effler, et al. , Physical Review Letters ,231107 (2019).[13] F. Acernese, M. Agathos, L. Aiello, A. Allocca, A. Am-ato, S. Ansoldi, S. Antier, M. Ar`ene, N. Arnaud, S. As- cenzi, et al. , Physical Review Letters , 231108 (2019).[14] S. L. Braunstein and P. van Loock, Reviews of ModernPhysics , 513 (2005).[15] S. Zeytino˘glu, A. ˙Imamo˘glu, and S. Huber, Physical Re-view X , 021041 (2017).[16] C. K. Hong and L. Mandel, Physical Review Letters ,323 (1985).[17] V. Buˇzek, A. Vidiella-Barranco, and P. L. Knight, Phys-ical Review A , 6570 (1992).[18] C. C. Gerry and E. E. Hach III, Physics Letters A ,185 (1993).[19] E. E. Hach III and C. C. Gerry, Physical Review A ,490 (1994).[20] K. Zhu, Q. Wang, and X. Li, JOSA B , 1287 (1993).[21] D. Stoler, B. E. A. Saleh, and M. C. Teich, Optica Acta:International Journal of Optics , 345 (1985).[22] A. Joshi and R. R. Puri, Journal of Modern Optics ,557 (1989).[23] B. Baseia, A. F. De Lima, and G. C. Marques, PhysicsLetters A , 1 (1995).[24] Y. A. Barbosa, G. C. Marques, and B. Baseia, PhysicaA: Statistical Mechanics and its Applications , 346(2000).[25] W. Schleich, M. Pernigo, and F. L. Kien, Physical Re-view A , 2172 (1991).[26] G. S. Agarwal and K. Tara, Physical Review A , 492(1991).[27] R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Physical Review Letters , 2409 (1985).[28] L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, PhysicalReview Letters , 2520 (1986).[29] U. L. Andersen, T. Gehring, C. Marquardt, andG. Leuchs, Physica Scripta , 053001 (2016).[30] A. Otterpohl, F. Sedlmeir, U. Vogl, T. Dirmeier,G. Shafiee, G. Schunk, D. V. Strekalov, H. G. L. Schwe-fel, T. Gehring, U. L. Andersen, et al. , Optica , 1375(2019).[31] Y. Gao, J. Feng, Y. Li, and K. Zhang, Laser PhysicsLetters , 055202 (2019).[32] Y. Zhao, Y. Okawachi, J. K. Jang, X. Ji, M. Lipson,and A. L. Gaeta, Physical Review Letters , 193601(2020).[33] C. Gerry, P. Knight, and P. L. Knight, Introductoryquantum optics (Cambridge university press, 2005).[34] L. O’Driscoll, R. Nichols, and P. A. Knott, QuantumMachine Intelligence , 5 (2019).[35] A. Zavatta, S. Viciani, and M. Bellini, Science , 660 (2004); Physical Review A , 023820 (2005).[36] S. Sivakumar, Physical Review A , 035802 (2011).[37] S. U. Shringarpure and J. D. Franson, Physical ReviewA , 043802 (2019).[38] C. M. Caves, Physical Review D , 1817 (1982); C. M.Caves and B. L. Schumaker, Physical Review A , 3068(1985).[39] M. Hillery, Physical Review A , 3796 (1987).[40] V. V. Dodonov, I. A. Malkin, and V. I. Man’Ko, Physica , 597 (1974).[41] E. Jones, T. Oliphant, P. Peterson, et al. , “SciPy: Opensource scientific tools for Python,” (2001–).[42] B. Yurke and D. Stoler, Physical Review Letters , 13(1986).[43] W. M. Charles, K. S. Thorne, J. A. Wheeler, et al. , SanFrandisco: WH Freeman and Company (1973).[44] C.-I. Kuo and L. H. Ford, Physical Review D , 4510(1993).[45] W. Schleich and J. A. Wheeler, JOSA B4