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Superposition of new phase states: generation and properties
Physics Letters A 337 (2005) 296–304 www.elsevier.com/locate/pla
Superposition of new phase states: generation and properties A. Aragão a,∗ , Paula B. Monteiro a , A.T. Avelar a,b , B. Baseia a a Instituto de Física, Universidade Federal de Goiás, 74.001-970, Goiânia (GO), Brazil b Instituto de Física, Universidade de Brasília, 70.919-970, Brasília (DF), Brazil
Received 7 December 2004; received in revised form 31 January 2005; accepted 2 February 2005
Communicated by P.R. Holland
Abstract In a recent paper [Phys. Lett. A 331 (2004) 366] we have introduced a new kind of phase state. Here we investigate properties of superposition of these states and their generation for trapped fields inside a high-Q cavity and for running fields. 2005 Elsevier B.V. All rights reserved. PACS: 42.50.Dv Keywords: Quantum state engineering; Partial phase states; Superposed phase state
1. Introduction In a previous paper [1] we have introduced a new kind of phase state, which differs from the traditional one, introduced in the literature by Pegg–Barnett [2]: while in [2] the ideal phase state emerges from a limit taken in the Hilbert space dimension: limN →∞ N n=0 , in [1] the ideal phase state is got from a limit implemented upon the arrangement of a coherent state |α (with |α| = R → ∞), plus a strong radial squeezing (r → ∞) in the phase space—with R and r obeying a prescription that r = ln R. As discussed in [1], this ideal phase state is nonphysical, since it requires in* Corresponding author.
E-mail address: [email protected] (A. Aragão). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.02.013
finite amount of energy for its creation [3]. However, reasonable approximations are not forbidden, hence it may turn to be useful for many applications, reminding us of the extensively used ideal plane waves. Generation of new states of quantized systems turned out to be an important subject in the last decade, concerning with both atomic and field states. In the later case the issue may refer to either trapped fields inside high-Q microwave cavities or running fields. The importance of preparing new states of quantized systems comes from their potential applications, many of them mentioned in [1]. As example of field states one may cite the number states [4], coherent states [5], squeezed states [6], and Schrödinger cat states [7]. The study of superposed states is also interesting, since they generally exhibit properties which are not present in the component states. To our knowledge this
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feature was first observed in Ref. [8], when studying superposition of two number states: it exhibits squeezing, an effect not shown by any number state. Another example: coherent states are most classical, but superposing two of them results a nonclassical state. Superpositions of Pegg–Barnett phase states [2], their preparation and properties were studied recently [9]. In the present Letter, for comparison, we will again study the generation and properties of superposed phase states (SPS), but now using as components of the superposition the new phase states introduced in [1]. This Letter is arranged as follows. In Section 2 we discuss the scheme preparing the SPS, for stationary and traveling waves. In the Section 3 we study nonclassical properties shown by this state. Section 4 treats its degree of nonclassicality and Section 5 presents the comments and conclusion.
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zaˆ †2 )/2]. The relations r = ln R and φ = 2θ − π are necessary prescriptions stabilized in [1]. Here we modify the first prescription to r = ln(R/λ), in order to improve the formalism. In fact we obtain, for 2 θ = θ + π : R, θ |R, θ = e−2λ , which vanishes for λ 1. For λ = 2 one obtains R, θ |R, θ 10−4 hence, λ 2 is sufficient to orthogonalize the states |R, θ and |R, θ + π. We will denote the SPPS as ± Φ (R, θ1 , θ2 ) = η± |R, θ2 ± |R, θ1 , (2) where −1/2 . η± = 2 ± 2Re R, θ1 |R, θ2
(3)
We have that, R, θ1 |R, θ2 =
√ −λ2 1−Aeiϕ + (R2 −λ2 )(1−Aei2ϕ ) R 2 +λ2 Ae ,
(4)
with
−1 A = 1 + 2B sin2 ϕ − iB sin 2ϕ ,
2. Generation of the SPS The phase state (PS) introduced in [1] is a limiting case of a partial phase state (PPS), then our preliminary strategy here will consist in the creation of superposition of partial phase states (SPPS), the SPS emerging as a convenient limit taken in the procedure. To this end we resume the PS studied in [1], |θ = lim α(R, θ ), z r(R), φ(θ ) = lim |R, θ , R→∞
R→∞
(1) ˆ ˆ where |α, z = D(α) S(z)|0, with α = Reiθ , z = reiφ , † ∗ ˆ ˆ ˆ and S(z) = exp[(z∗ aˆ 2 − D(α) = exp(α aˆ − α a)
(5)
2 )2 /4λ2 R 2 and ϕ = θ − θ rewhere B = (R 2 − λ√ 2 1 ± sulting = η → 1/ 2 in the limit R → ∞, since R, θ1 |R, θ2 = 0 for ϕ = 0. So, the SPS is got from the limit ± Ψ = lim Φ ± (R, θ1 , θ2 ) = η± |θ2 ± |θ1 . (6) R→∞
2.1. Stationary fields The present procedure mimics that of Ref. [9], using an experimental arrangement shown in Fig. 1, proposed in Ref. [7] for other purposes. To create the
Fig. 1. Experimental setup preparing of SPPS in a high-Q cavity using dispersive atomic probes.
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stationary SPPS we start from an initial field previously prepared in a convenient type of PPS defined in Ref. [1]. This initial field then interacts dispersively with a two-level (Rydberg) atom traversing the cavity. The SPS is obtained from a selective atomic detection which projects an entangled state describing the entire atom-field system. To be reliable the method requires the use of a high-Q microwave cavity (to trap the field inside it and reducing decoherence effects), at low temperatures (T ∼ 4 K, to reduce the presence of thermal photons), and the use of atomic ionization detectors having high resolution (to improve the success probability and fidelity of the state being prepared). The cavity is placed between two low-Q microwave cavities (Ramsey zones), the first of them preparing arbitrary superposition of excited and ground atomic states (|e and |g), the second one producing atomfield entangled states. In Fig. 1 Rydberg atoms, ejected from a source Sa with selected velocities, are prepared in circular excited state |e (principal quantum number n = 51 for rubidium) by an appropriated laser beam. When the atoms cross both Ramsey zones R1 and R2 they lead the state |e (|g) to |g + |e (|g − |e). Inside the cavity occurs a dispersive atom-field interaction, described by the effective Hamiltonian [10], † Hˆ int = hω ¯ eff aˆ aˆ |ii| − |ee| ,
(7)
with ωeff = 2d2 /δ, d being the atomic dipole moment and δ standing for detuning between the field frequency and atomic transition frequency, from the
state |e to an auxiliary state |i. Thus the atom crossing the cavity produces a phase-shift in the field state when it is in the excited state |e, but no phase-shift appears when the atom is in the ground state |g. Next consider the field inside the cavity being initially prepared in a PPS, |R, θ1 . As the atom crosses the system, the evolution of the (entangled) atom-field state |ΨAF is given in the following steps: (i) atom before R1 → |ΨAF = |e|R, θ1 ; (ii) atom after R1 → |ΨAF = (|g + |e)|R, θ1 ; (iii) atom after the cavity → |ΨAF = |g|R, θ1 + |e|R, ϕ + θ1 ; (iv) atom after R2 → |ΨAF = |g(|R, ϕ +θ1 +|R, θ1 )+|e(|R, ϕ + θ1 − |R, θ1 ). Now, taking θ2 = θ1 + ϕ, the detection in |g or |e leads the cavity field to the state |Φ ± (R, θ1 , θ2 ), as given in Eq. (2). Finally, when this procedure uses the SPPS with R → ∞ one gets our desired SPS given in Eq. (6). 2.2. Traveling fields Our SPS can also be produced for running waves of the electromagnetic field. The experimental setup illustrated in Fig. 2 was proposed in Ref. [11] to produce Schrödinger cat states. The present procedure mimics that in Ref. [12] by substituting the Pegg–Barnett PS by the new PS introduced in [1]. The scheme consists of a Mach–Zehnder interferometer (MZI) including a Kerr-medium, which couples b-mode to an external a-mode. Initially a single photon (mode b) and vac-
Fig. 2. Schematic illustration of the setup preparing the SPPS as running waves. K stands for Kerr-medium.
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uum state (mode c) enter the first ideal 50/50 symmetric beam splitter (BS1). The action of BS1 leads the foregoing initial state describing the whole system to (|1b |0c + i|0b |1c )|R, θ1 a , where |R, θ1 a is the initial PPS in the mode a. The dispersive (Kerr) interaction of a- and b-modes is described by the Hamiltonian Hˆ K = hK ¯ aˆ † aˆ bˆ † bˆ [13], where K is proportional to the third order nonlinear susceptibility of the medium. As result of the action of the Kerr-medium the state describing the entire system is |1b |0c |R, θ1 − ϕa + i|0b |1c |R, θ1 a , where ϕ = Kl/v, l is the length of the Kerr-medium and v is the velocity of light in it. Now, after BS2 the (entangled) state of the system reads |Ψ ac =
1 |1b |0c |R, θ1 − ϕa − |R, θ1 a 2 + i|0b |1c |R, θ1 − ϕa + |R, θ1 a .
(8)
If detector D1 (D2) fires, signalizing the detection of the state |1b |0c (|0b |1c ), the a-mode is projected onto the state |Φ − (R, θ1 , θ1 − ϕ) (|Φ + (R, θ1 , θ1 − ϕ)). Thus, taking θ2 = θ1 − ϕ, the detection D1 (D2) leads the a-mode to the SPPS |Φ + (R, θ1 , θ2 ) (|Φ − (R, θ1 , θ2 )) given in Eq. (2). As achieved in the stationary case, we obtain our SPS from the SPPS in the limit R → ∞.
3. Nonclassical properties As before, we first investigate the SPPS and obtain the nonclassical properties of the SPS from those of the SPPS in limit R → ∞. 3.1. Photon statistics The coefficients of the SPPS expanded in the ± Fock’s basis, |Φ ± (R, θ1 , θ2 ) = ∞ n=0 Cn |n, given by,
−λ2 R 2 2Rλ R 2 − λ2 n ± ± 2 +λ2 R −i e Cn = η R 2 + λ2 2(R 2 + λ2 ) inθ2 (e 2R 2 ± einθ1 ) 2 , × Hn iλ (9) √ R 4 − λ4 n!
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furnishes its photon number distribution, Pn± = |Cn± |2 . We obtain, −2λ2 R 2
Pn±
|η± |2 Rλ[1 ± cos(nϕ)]e R2 +λ2 = 2n−2 n!(R 2 + λ2 ) 2 2 2R 2 λ − R2 n 2 iλ H . × n λ2 + R 2 R 4 − λ4
(10)
As occurs for each PPS-component, Pn± → 0 when R → ∞. Note that even for finite values of R, we obtain Pn+ = 0 (Pn− = 0) whenever nϕ = (2k + 1)π (nϕ = 2kπ ), k = 1, 2, 3, . . . . For ϕ = π , Pn+ (Pn− ) results an even (odd) function of n. When ϕ = π/2 we find Pn+ = 0 for n = 2, 6, 10, . . . and Pn− = 0 for n = 0, 4, 8, . . . . When ϕ = π/3 we find Pn+ = 0 for n = 3, 9, 15, . . . and Pn− = 0 for n = 0, 6, 12, . . . , etc. Hence, for N integer and setting ϕ = π/N , the distance between the zeros of Pn± increases when N grows. This distance is given by D = 2N , N = π/ϕ, integer. Now, since the limit R → ∞ does not destroy these characteristics, they are preserved in the SPS. Fig. 3 exhibits the plots of the photon number distribution Pn+ , for different values of the parameter R, whereas Fig. 4 is the same for different values of parameter ϕ. In Fig. 3 we see that Pn+ is attenuated when R increases, as expected, since this is a feature shown by each PPS-component. For large values of n the statistics of the SPPS becomes almost constant, the same result characterizing their components. Similar results were obtained superposing the Pegg–Barnett phase states [9]. In Fig. 4 the SPPS has no parity, since ϕ = π . In this case, contrary to Fig. 3, the statistical distribution has an envelop which oscillates, being no longer a monotonic function. Oscillations of the statistical distribution were shown to be connected with interference in the phase space [14]. Hence interference effects obtained with SPPS having a definite parity (i.e., when ϕ = π ) are more regular than those obtained with SPPS having no parity. The mentioned oscillations in the statistical distribution are built-in Eq. (10), but they become transparent when we write this equation in the equivalent form, 2 2 Pn± = 2η± n|R, θ1 1 ± cos (nϕ) . (11) We have also observed that all SPPS are superPoissonian (Mandel factor Q > 0, Q = ( nˆ 2 − n)/ ˆ n), ˆ an expected result, since the PPS-compo-
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Fig. 3. Plots of Pn+ versus n for ϕ = π/4, λ = 1 and: (a) R = 10; and (b) R = 20.
Fig. 4. Plots of Pn± versus n, for R = 10, λ = 1 and (a) ϕ = π/4, for Pn+ ; (b) ϕ = π/3, for Pn− .
nents are always super-Poissonian. In the limit R → ∞ the state becomes most super-Poissonian, i.e., Q → ∞, a result also shown by the Pegg–Barnett phase states [2] and their superpositions [9] (differently, in [2] the limit is taken in a N -dimensional Hilbert space, N → ∞). 3.2. Collapses and revivals of atomic inversion In experiments involving electromagnetic cavities, one monitors the population of atomic states as function of time [15]. When a two-level (Rydberg) atom interacts with a (single mode) field, with the whole system being previously prepared in a given state, the time evolution of the atom-field system is described by the Jaynes–Cummings Hamiltonian [16]. Assuming resonance between the field and atom, plus the rotating
wave approximation,1 this Hamiltonian is written as Hˆ = h¯ w aˆ † aˆ + h¯ w σˆ z /2 + h¯ λ σˆ + aˆ + σˆ − aˆ † , (12) where aˆ † (a) ˆ is the creation (annihilation) operator for photons; σˆ z = (|ee| − |gg|) is the atomic inversion operator, with |e (|g) standing for the atomic excited (ground) state; σˆ + (σˆ − ) is the raising (lowering) operator for the atom; w is the field (and atomic) frequency and λ stands for the atom-field coupling constant. The atomic inversion is obtained via [17]: W (T ) = ΨAF (0)|σˆ z (T )|ΨAF (0), with T = λt. In this expression |ΨAF (0) stands for the initial state describing the whole atom-field system. Here we take |ΨAF (0) = |ΨA (0)|ΨF (0) with |ΨA (0) = |e and 1 This corresponds to neglecting the coupling term hλ(σˆ + aˆ † + ¯ − σˆ a). ˆ
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Fig. 5. Plots of atomic inversion W (T ) versus time T = λt for (a) (+) λ = 1, R = 2, ϕ = π ; (b) (−) λ = 1, R = 2, ϕ = 3π/25; (c) (+) R = 10, λ = 1, ϕ = π/25; (d) (+) λ = 2, R = 10, ϕ = π . The signals (+) and (−) stand for the even and odd superpositions.
|Ψ Then we find that: W (T ) = √ F∞(0) ±as our SPPS. ± n=0 Pn cos(2T n + 1), with Pn given in Eq. (10). Fig. 5 shows plots of the atomic inversion W (T ) as function of time T = λt. We note that for certain values of the phase ϕ, we see the SPPS exhibiting the collapse-revival effect (cf. Fig. 5(b) and (d)). The same effect was observed in Ref. [9] using superpositions of Pegg–Barnett PS. When R becomes large (cf. Fig. 5(c) and (d)) we observe attenuation of the atomic inversion, which is an expected result since their PPS-components tend to the PS displaying this feature. This attenuation reduces the atomic inversion to zero when R → ∞ (not shown in figures), which means a trapping of the atomic inversion; the same occurring with our PS [1] and the Pegg–Barnett PS [2]. 3.3. Quasi-distributions Using the characteristic function χ(η) for a pure ˆ state |Ψ , χ(η) = Ψ |D(η)|Ψ , we obtain a general
class of quasi-distributions [18,19], written as w 1 2 ∗ ∗ F (ζ ; w) = 2 e− 2 |η| +η ζ −ηζ χ(η) d 2 η. π
(13)
all
For w = 0, F (ζ ; 0) furnishes the Wigner distribution, whereas w = 1 yields the Husimi Q-function. For our SPPS Eq. (13) results, with ζ = x + iy, 2 F (x, y; w) = η± /π 2 h(θ1 , θ1 ) + h(θ2 , θ2 ) ± h(θ1 , θ2 ) + h(θ2 , θ1 ) , (14) where, h(θk , θk ) w 2 ∗ ∗ ˆ θk d 2 η. = e− 2 |η| +η ζ −ηζ R, θk |D(η)|R, (15) Fig. 6 shows the Wigner function for some values of the phase difference ϕ = θ2 − θ1 . These plots turn
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Fig. 6. Plots of Wigner function for |Φ + (R, θ1 , θ2 ), with R = 5, λ = 1 and growing values of ϕ = θ2 − θ1 : (a) (θ1 , θ2 ) = (2π/16, 6π/16); (b) (θ1 , θ2 ) = (0, π/2).
Fig. 7. Plots of y = P (Θ) versus x = Θ for (a) (+) λ = 1, (θ1 , θ2 ) = (0, π/4) and R = 3 and (b) (−) λ = 1, (θ1 , θ2 ) = (−π/2, π/2) and R = 15.
out transparent the presence of the PPS-components, their interference lying between them—a feature not appearing in Ref. [9] using superposition of Pegg– Barnett phase states. The Husimi Q-function, not shown in figures, allows us to investigate the nonclassical depth of the SPPS and SPS, as implemented in Section 4. 3.4. Phase distribution There are various theoretical approaches describing the phase observable. Some of them are motivated by the aim of expressing the phase as the complement of the photon number, in the spirit of Dirac’s original work [20]. Although these approaches are quite
distinct, they lead to the same phase probability distribution P (Θ) for a field in state |Ψ , ∞ 2 1 n|Ψ e−inΘ . P (Θ) = (16) 2π n=0
It is worth noting that the simplest way of obtaining this probability distribution is by taking the overlap between |Ψ and the Pegg–Barnett phase state in the appropriate limit. Next, replacing in (16) the Cn± = n|Φ ± (R, θ1 , θ2 ) given in (9) we find the phase distribution for our SPPS −2λ2 R 2
2 |η± |2 Rλe R2 +λ2 ± g (Θ) , P (Θ) = 2 2 π(R + λ ) ±
(17)
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where ∞ ± −i g (Θ) =
303
R 2 − λ2 n 2(R 2 + λ2 ) n=0 2R 2 2 × Hn iλ R 4 − λ4
√ × ein(θ1 −Θ) einϕ/2 f ± (ϕ)/ n!,
f + (ϕ) = 2 cos(nϕ/2),
(18)
f − (ϕ) = 2i sin(nϕ/2).
where Fig. 7 shows plots of: (a) P + (Θ) for λ = 1, (θ1 , θ2 ) = (0, π/4), R = 3; (b) P − (Θ) for λ = 1, (θ1 , θ2 ) = (−π/2, π/2), R = 15. We note the distributions concentrated around Θ = θ1 and Θ = θ2 . When R grows interference disappears and P ± (Θ) tend to a Dirac-type distribution δ(Θ − θ1 ) + δ(Θ − θ2 ). 4. Nonclassical depth A pertinent question in quantum optics is: “given two states exhibiting distinct nonclassical effects, which of them is more nonclassical than the other?”. This question has been addressed by various authors in the literature [21]. More recently [22], inspired on [23], the issue has been considered analyzing both phase-space and distance-type measures of nonclassicality. In [22] the criterium to quantify the nonclassical depth of a state |Ψ is taken as the minimum of distance of this state and that of the nearest classical (coherent) state. A straightforward procedure using this concept leads to the operational formula for the nonclassical degree, dm (Ψ ) = 1 − πQmax (Ψ ), where Qmax (Ψ ) stands for maximum of Husimi Q-function. For our SPPS, using Eq. (13) for w = 1, we find the result shown in Fig. 8. Note that the nonclassical degree of this state is small (dm ∼ 0.1) for R ∼ 1 and tends to its maximum value (dm = 1) for R → ∞, when the SPPS tends to the SPS. The same limit (dm = 1) has been obtained for the Pegg–Barnett PS [24] and our PS introduced in [1]. The influence of phase ϕ upon the degree of nonclassicality is very small (not shown in figures). 5. Comments and conclusion We have studied preparation of SPS for stationary (Section 2.1) and running fields (Section 2.2), its nonclassical properties (Section 3) and nonclassical depth
Fig. 8. Plot of nonclassical depth y = dm (|Φ + (R, θ1 , θ2 )) versus R, for λ = 1 and ϕ = π/4.
(Section 4). The results of the SPS are obtained from those of SPPS in the limit of large values of R = |α| and r = |z| plus the prescription r = ln(R/λ), λ 1 (cf. Section 2). As discussed in [1], the limit R → ∞ furnishes an ideal PS since reaching this limit involves infinite energy. However, convenient approximations of the ideal case may be useful in certain circumstances, in the same way as plane waves become useful in many situations. We have shown that the SPPS (and SPS) exhibits various nonclassical properties: oscillations in the statistical distribution Pn± (Section 3.1)— which are connected with interference in the phase space—and collapse and revival effect of the atomic inversion W (T ) (Section 3.2). These two effects diminish when R grows and vanish if R → ∞, when each component of the superposition becomes a PS having these characteristics. We have also obtained the Wigner distribution and the Husimi Q-function (Section 3.3): the first showing in which way the interference terms emerge in the scenario (cf. Fig. 6), exhibiting both components of the superposition and the interference terms, a result not observed in [9]. Comparison of our Wigner functions with those in [9] is difficult due to the absence of corresponding parameters in these schemes. Concerning with the Husimi Q-function it allows us to calculate the measure of nonclassicality using the criterium discussed in [22] (Section 4). It was shown that our superposed state
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becomes more nonclassical when R grows, and most nonclassical (dm = 1) in the limit R → ∞, which coincides with the result found in [9] using the Pegg– Barnett SPS. For completeness, Section 3.4 studied the phase distribution of the SPS. In principle the present procedure can be pursued further, by replicating the experimental setup. By passing a sequence of atoms with controlled velocities throughout the cavity we obtain: a superposition having four PPS-components when using two atoms; eight PPS-components when using three atoms; etc. In this case the output field state obtained from the j th atom works as the input field state for the next one. This procedure was applied to create “high-generation Schrödinger-cat states” inside a cavity [25]. This extension is also valid for traveling waves by replicating the MZI and considering the output state of the j th MZI as input state of the next one. Of course, decoherence effects imply serious restriction to these replications, at least in the present status of technology. Here, we have restricted ourselves to the simple case of SPPS and SPS containing two components, as given in Eq. (6).
Acknowledgements We thank Professor N. Zagury for discussions (A.A.), CAPES (A.A.) and CNPq (P.B.M., A.T.A., B.B.), Brazilian agencies, for partially supporting this work.
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Superposition of new phase states: generation and properties A. Aragão a,∗ , Paula B. Monteiro a , A.T. Avelar a,b , B. Baseia a a Instituto de Física, Universidade Federal de Goiás, 74.001-970, Goiânia (GO), Brazil b Instituto de Física, Universidade de Brasília, 70.919-970, Brasília (DF), Brazil
Received 7 December 2004; received in revised form 31 January 2005; accepted 2 February 2005
Communicated by P.R. Holland
Abstract In a recent paper [Phys. Lett. A 331 (2004) 366] we have introduced a new kind of phase state. Here we investigate properties of superposition of these states and their generation for trapped fields inside a high-Q cavity and for running fields. 2005 Elsevier B.V. All rights reserved. PACS: 42.50.Dv Keywords: Quantum state engineering; Partial phase states; Superposed phase state
1. Introduction In a previous paper [1] we have introduced a new kind of phase state, which differs from the traditional one, introduced in the literature by Pegg–Barnett [2]: while in [2] the ideal phase state emerges from a limit taken in the Hilbert space dimension: limN →∞ N n=0 , in [1] the ideal phase state is got from a limit implemented upon the arrangement of a coherent state |α (with |α| = R → ∞), plus a strong radial squeezing (r → ∞) in the phase space—with R and r obeying a prescription that r = ln R. As discussed in [1], this ideal phase state is nonphysical, since it requires in* Corresponding author.
E-mail address: [email protected] (A. Aragão). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.02.013
finite amount of energy for its creation [3]. However, reasonable approximations are not forbidden, hence it may turn to be useful for many applications, reminding us of the extensively used ideal plane waves. Generation of new states of quantized systems turned out to be an important subject in the last decade, concerning with both atomic and field states. In the later case the issue may refer to either trapped fields inside high-Q microwave cavities or running fields. The importance of preparing new states of quantized systems comes from their potential applications, many of them mentioned in [1]. As example of field states one may cite the number states [4], coherent states [5], squeezed states [6], and Schrödinger cat states [7]. The study of superposed states is also interesting, since they generally exhibit properties which are not present in the component states. To our knowledge this
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feature was first observed in Ref. [8], when studying superposition of two number states: it exhibits squeezing, an effect not shown by any number state. Another example: coherent states are most classical, but superposing two of them results a nonclassical state. Superpositions of Pegg–Barnett phase states [2], their preparation and properties were studied recently [9]. In the present Letter, for comparison, we will again study the generation and properties of superposed phase states (SPS), but now using as components of the superposition the new phase states introduced in [1]. This Letter is arranged as follows. In Section 2 we discuss the scheme preparing the SPS, for stationary and traveling waves. In the Section 3 we study nonclassical properties shown by this state. Section 4 treats its degree of nonclassicality and Section 5 presents the comments and conclusion.
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zaˆ †2 )/2]. The relations r = ln R and φ = 2θ − π are necessary prescriptions stabilized in [1]. Here we modify the first prescription to r = ln(R/λ), in order to improve the formalism. In fact we obtain, for 2 θ = θ + π : R, θ |R, θ = e−2λ , which vanishes for λ 1. For λ = 2 one obtains R, θ |R, θ 10−4 hence, λ 2 is sufficient to orthogonalize the states |R, θ and |R, θ + π. We will denote the SPPS as ± Φ (R, θ1 , θ2 ) = η± |R, θ2 ± |R, θ1 , (2) where −1/2 . η± = 2 ± 2Re R, θ1 |R, θ2
(3)
We have that, R, θ1 |R, θ2 =
√ −λ2 1−Aeiϕ + (R2 −λ2 )(1−Aei2ϕ ) R 2 +λ2 Ae ,
(4)
with
−1 A = 1 + 2B sin2 ϕ − iB sin 2ϕ ,
2. Generation of the SPS The phase state (PS) introduced in [1] is a limiting case of a partial phase state (PPS), then our preliminary strategy here will consist in the creation of superposition of partial phase states (SPPS), the SPS emerging as a convenient limit taken in the procedure. To this end we resume the PS studied in [1], |θ = lim α(R, θ ), z r(R), φ(θ ) = lim |R, θ , R→∞
R→∞
(1) ˆ ˆ where |α, z = D(α) S(z)|0, with α = Reiθ , z = reiφ , † ∗ ˆ ˆ ˆ and S(z) = exp[(z∗ aˆ 2 − D(α) = exp(α aˆ − α a)
(5)
2 )2 /4λ2 R 2 and ϕ = θ − θ rewhere B = (R 2 − λ√ 2 1 ± sulting = η → 1/ 2 in the limit R → ∞, since R, θ1 |R, θ2 = 0 for ϕ = 0. So, the SPS is got from the limit ± Ψ = lim Φ ± (R, θ1 , θ2 ) = η± |θ2 ± |θ1 . (6) R→∞
2.1. Stationary fields The present procedure mimics that of Ref. [9], using an experimental arrangement shown in Fig. 1, proposed in Ref. [7] for other purposes. To create the
Fig. 1. Experimental setup preparing of SPPS in a high-Q cavity using dispersive atomic probes.
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stationary SPPS we start from an initial field previously prepared in a convenient type of PPS defined in Ref. [1]. This initial field then interacts dispersively with a two-level (Rydberg) atom traversing the cavity. The SPS is obtained from a selective atomic detection which projects an entangled state describing the entire atom-field system. To be reliable the method requires the use of a high-Q microwave cavity (to trap the field inside it and reducing decoherence effects), at low temperatures (T ∼ 4 K, to reduce the presence of thermal photons), and the use of atomic ionization detectors having high resolution (to improve the success probability and fidelity of the state being prepared). The cavity is placed between two low-Q microwave cavities (Ramsey zones), the first of them preparing arbitrary superposition of excited and ground atomic states (|e and |g), the second one producing atomfield entangled states. In Fig. 1 Rydberg atoms, ejected from a source Sa with selected velocities, are prepared in circular excited state |e (principal quantum number n = 51 for rubidium) by an appropriated laser beam. When the atoms cross both Ramsey zones R1 and R2 they lead the state |e (|g) to |g + |e (|g − |e). Inside the cavity occurs a dispersive atom-field interaction, described by the effective Hamiltonian [10], † Hˆ int = hω ¯ eff aˆ aˆ |ii| − |ee| ,
(7)
with ωeff = 2d2 /δ, d being the atomic dipole moment and δ standing for detuning between the field frequency and atomic transition frequency, from the
state |e to an auxiliary state |i. Thus the atom crossing the cavity produces a phase-shift in the field state when it is in the excited state |e, but no phase-shift appears when the atom is in the ground state |g. Next consider the field inside the cavity being initially prepared in a PPS, |R, θ1 . As the atom crosses the system, the evolution of the (entangled) atom-field state |ΨAF is given in the following steps: (i) atom before R1 → |ΨAF = |e|R, θ1 ; (ii) atom after R1 → |ΨAF = (|g + |e)|R, θ1 ; (iii) atom after the cavity → |ΨAF = |g|R, θ1 + |e|R, ϕ + θ1 ; (iv) atom after R2 → |ΨAF = |g(|R, ϕ +θ1 +|R, θ1 )+|e(|R, ϕ + θ1 − |R, θ1 ). Now, taking θ2 = θ1 + ϕ, the detection in |g or |e leads the cavity field to the state |Φ ± (R, θ1 , θ2 ), as given in Eq. (2). Finally, when this procedure uses the SPPS with R → ∞ one gets our desired SPS given in Eq. (6). 2.2. Traveling fields Our SPS can also be produced for running waves of the electromagnetic field. The experimental setup illustrated in Fig. 2 was proposed in Ref. [11] to produce Schrödinger cat states. The present procedure mimics that in Ref. [12] by substituting the Pegg–Barnett PS by the new PS introduced in [1]. The scheme consists of a Mach–Zehnder interferometer (MZI) including a Kerr-medium, which couples b-mode to an external a-mode. Initially a single photon (mode b) and vac-
Fig. 2. Schematic illustration of the setup preparing the SPPS as running waves. K stands for Kerr-medium.
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uum state (mode c) enter the first ideal 50/50 symmetric beam splitter (BS1). The action of BS1 leads the foregoing initial state describing the whole system to (|1b |0c + i|0b |1c )|R, θ1 a , where |R, θ1 a is the initial PPS in the mode a. The dispersive (Kerr) interaction of a- and b-modes is described by the Hamiltonian Hˆ K = hK ¯ aˆ † aˆ bˆ † bˆ [13], where K is proportional to the third order nonlinear susceptibility of the medium. As result of the action of the Kerr-medium the state describing the entire system is |1b |0c |R, θ1 − ϕa + i|0b |1c |R, θ1 a , where ϕ = Kl/v, l is the length of the Kerr-medium and v is the velocity of light in it. Now, after BS2 the (entangled) state of the system reads |Ψ ac =
1 |1b |0c |R, θ1 − ϕa − |R, θ1 a 2 + i|0b |1c |R, θ1 − ϕa + |R, θ1 a .
(8)
If detector D1 (D2) fires, signalizing the detection of the state |1b |0c (|0b |1c ), the a-mode is projected onto the state |Φ − (R, θ1 , θ1 − ϕ) (|Φ + (R, θ1 , θ1 − ϕ)). Thus, taking θ2 = θ1 − ϕ, the detection D1 (D2) leads the a-mode to the SPPS |Φ + (R, θ1 , θ2 ) (|Φ − (R, θ1 , θ2 )) given in Eq. (2). As achieved in the stationary case, we obtain our SPS from the SPPS in the limit R → ∞.
3. Nonclassical properties As before, we first investigate the SPPS and obtain the nonclassical properties of the SPS from those of the SPPS in limit R → ∞. 3.1. Photon statistics The coefficients of the SPPS expanded in the ± Fock’s basis, |Φ ± (R, θ1 , θ2 ) = ∞ n=0 Cn |n, given by,
−λ2 R 2 2Rλ R 2 − λ2 n ± ± 2 +λ2 R −i e Cn = η R 2 + λ2 2(R 2 + λ2 ) inθ2 (e 2R 2 ± einθ1 ) 2 , × Hn iλ (9) √ R 4 − λ4 n!
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furnishes its photon number distribution, Pn± = |Cn± |2 . We obtain, −2λ2 R 2
Pn±
|η± |2 Rλ[1 ± cos(nϕ)]e R2 +λ2 = 2n−2 n!(R 2 + λ2 ) 2 2 2R 2 λ − R2 n 2 iλ H . × n λ2 + R 2 R 4 − λ4
(10)
As occurs for each PPS-component, Pn± → 0 when R → ∞. Note that even for finite values of R, we obtain Pn+ = 0 (Pn− = 0) whenever nϕ = (2k + 1)π (nϕ = 2kπ ), k = 1, 2, 3, . . . . For ϕ = π , Pn+ (Pn− ) results an even (odd) function of n. When ϕ = π/2 we find Pn+ = 0 for n = 2, 6, 10, . . . and Pn− = 0 for n = 0, 4, 8, . . . . When ϕ = π/3 we find Pn+ = 0 for n = 3, 9, 15, . . . and Pn− = 0 for n = 0, 6, 12, . . . , etc. Hence, for N integer and setting ϕ = π/N , the distance between the zeros of Pn± increases when N grows. This distance is given by D = 2N , N = π/ϕ, integer. Now, since the limit R → ∞ does not destroy these characteristics, they are preserved in the SPS. Fig. 3 exhibits the plots of the photon number distribution Pn+ , for different values of the parameter R, whereas Fig. 4 is the same for different values of parameter ϕ. In Fig. 3 we see that Pn+ is attenuated when R increases, as expected, since this is a feature shown by each PPS-component. For large values of n the statistics of the SPPS becomes almost constant, the same result characterizing their components. Similar results were obtained superposing the Pegg–Barnett phase states [9]. In Fig. 4 the SPPS has no parity, since ϕ = π . In this case, contrary to Fig. 3, the statistical distribution has an envelop which oscillates, being no longer a monotonic function. Oscillations of the statistical distribution were shown to be connected with interference in the phase space [14]. Hence interference effects obtained with SPPS having a definite parity (i.e., when ϕ = π ) are more regular than those obtained with SPPS having no parity. The mentioned oscillations in the statistical distribution are built-in Eq. (10), but they become transparent when we write this equation in the equivalent form, 2 2 Pn± = 2η± n|R, θ1 1 ± cos (nϕ) . (11) We have also observed that all SPPS are superPoissonian (Mandel factor Q > 0, Q = ( nˆ 2 − n)/ ˆ n), ˆ an expected result, since the PPS-compo-
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Fig. 3. Plots of Pn+ versus n for ϕ = π/4, λ = 1 and: (a) R = 10; and (b) R = 20.
Fig. 4. Plots of Pn± versus n, for R = 10, λ = 1 and (a) ϕ = π/4, for Pn+ ; (b) ϕ = π/3, for Pn− .
nents are always super-Poissonian. In the limit R → ∞ the state becomes most super-Poissonian, i.e., Q → ∞, a result also shown by the Pegg–Barnett phase states [2] and their superpositions [9] (differently, in [2] the limit is taken in a N -dimensional Hilbert space, N → ∞). 3.2. Collapses and revivals of atomic inversion In experiments involving electromagnetic cavities, one monitors the population of atomic states as function of time [15]. When a two-level (Rydberg) atom interacts with a (single mode) field, with the whole system being previously prepared in a given state, the time evolution of the atom-field system is described by the Jaynes–Cummings Hamiltonian [16]. Assuming resonance between the field and atom, plus the rotating
wave approximation,1 this Hamiltonian is written as Hˆ = h¯ w aˆ † aˆ + h¯ w σˆ z /2 + h¯ λ σˆ + aˆ + σˆ − aˆ † , (12) where aˆ † (a) ˆ is the creation (annihilation) operator for photons; σˆ z = (|ee| − |gg|) is the atomic inversion operator, with |e (|g) standing for the atomic excited (ground) state; σˆ + (σˆ − ) is the raising (lowering) operator for the atom; w is the field (and atomic) frequency and λ stands for the atom-field coupling constant. The atomic inversion is obtained via [17]: W (T ) = ΨAF (0)|σˆ z (T )|ΨAF (0), with T = λt. In this expression |ΨAF (0) stands for the initial state describing the whole atom-field system. Here we take |ΨAF (0) = |ΨA (0)|ΨF (0) with |ΨA (0) = |e and 1 This corresponds to neglecting the coupling term hλ(σˆ + aˆ † + ¯ − σˆ a). ˆ
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Fig. 5. Plots of atomic inversion W (T ) versus time T = λt for (a) (+) λ = 1, R = 2, ϕ = π ; (b) (−) λ = 1, R = 2, ϕ = 3π/25; (c) (+) R = 10, λ = 1, ϕ = π/25; (d) (+) λ = 2, R = 10, ϕ = π . The signals (+) and (−) stand for the even and odd superpositions.
|Ψ Then we find that: W (T ) = √ F∞(0) ±as our SPPS. ± n=0 Pn cos(2T n + 1), with Pn given in Eq. (10). Fig. 5 shows plots of the atomic inversion W (T ) as function of time T = λt. We note that for certain values of the phase ϕ, we see the SPPS exhibiting the collapse-revival effect (cf. Fig. 5(b) and (d)). The same effect was observed in Ref. [9] using superpositions of Pegg–Barnett PS. When R becomes large (cf. Fig. 5(c) and (d)) we observe attenuation of the atomic inversion, which is an expected result since their PPS-components tend to the PS displaying this feature. This attenuation reduces the atomic inversion to zero when R → ∞ (not shown in figures), which means a trapping of the atomic inversion; the same occurring with our PS [1] and the Pegg–Barnett PS [2]. 3.3. Quasi-distributions Using the characteristic function χ(η) for a pure ˆ state |Ψ , χ(η) = Ψ |D(η)|Ψ , we obtain a general
class of quasi-distributions [18,19], written as w 1 2 ∗ ∗ F (ζ ; w) = 2 e− 2 |η| +η ζ −ηζ χ(η) d 2 η. π
(13)
all
For w = 0, F (ζ ; 0) furnishes the Wigner distribution, whereas w = 1 yields the Husimi Q-function. For our SPPS Eq. (13) results, with ζ = x + iy, 2 F (x, y; w) = η± /π 2 h(θ1 , θ1 ) + h(θ2 , θ2 ) ± h(θ1 , θ2 ) + h(θ2 , θ1 ) , (14) where, h(θk , θk ) w 2 ∗ ∗ ˆ θk d 2 η. = e− 2 |η| +η ζ −ηζ R, θk |D(η)|R, (15) Fig. 6 shows the Wigner function for some values of the phase difference ϕ = θ2 − θ1 . These plots turn
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Fig. 6. Plots of Wigner function for |Φ + (R, θ1 , θ2 ), with R = 5, λ = 1 and growing values of ϕ = θ2 − θ1 : (a) (θ1 , θ2 ) = (2π/16, 6π/16); (b) (θ1 , θ2 ) = (0, π/2).
Fig. 7. Plots of y = P (Θ) versus x = Θ for (a) (+) λ = 1, (θ1 , θ2 ) = (0, π/4) and R = 3 and (b) (−) λ = 1, (θ1 , θ2 ) = (−π/2, π/2) and R = 15.
out transparent the presence of the PPS-components, their interference lying between them—a feature not appearing in Ref. [9] using superposition of Pegg– Barnett phase states. The Husimi Q-function, not shown in figures, allows us to investigate the nonclassical depth of the SPPS and SPS, as implemented in Section 4. 3.4. Phase distribution There are various theoretical approaches describing the phase observable. Some of them are motivated by the aim of expressing the phase as the complement of the photon number, in the spirit of Dirac’s original work [20]. Although these approaches are quite
distinct, they lead to the same phase probability distribution P (Θ) for a field in state |Ψ , ∞ 2 1 n|Ψ e−inΘ . P (Θ) = (16) 2π n=0
It is worth noting that the simplest way of obtaining this probability distribution is by taking the overlap between |Ψ and the Pegg–Barnett phase state in the appropriate limit. Next, replacing in (16) the Cn± = n|Φ ± (R, θ1 , θ2 ) given in (9) we find the phase distribution for our SPPS −2λ2 R 2
2 |η± |2 Rλe R2 +λ2 ± g (Θ) , P (Θ) = 2 2 π(R + λ ) ±
(17)
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where ∞ ± −i g (Θ) =
303
R 2 − λ2 n 2(R 2 + λ2 ) n=0 2R 2 2 × Hn iλ R 4 − λ4
√ × ein(θ1 −Θ) einϕ/2 f ± (ϕ)/ n!,
f + (ϕ) = 2 cos(nϕ/2),
(18)
f − (ϕ) = 2i sin(nϕ/2).
where Fig. 7 shows plots of: (a) P + (Θ) for λ = 1, (θ1 , θ2 ) = (0, π/4), R = 3; (b) P − (Θ) for λ = 1, (θ1 , θ2 ) = (−π/2, π/2), R = 15. We note the distributions concentrated around Θ = θ1 and Θ = θ2 . When R grows interference disappears and P ± (Θ) tend to a Dirac-type distribution δ(Θ − θ1 ) + δ(Θ − θ2 ). 4. Nonclassical depth A pertinent question in quantum optics is: “given two states exhibiting distinct nonclassical effects, which of them is more nonclassical than the other?”. This question has been addressed by various authors in the literature [21]. More recently [22], inspired on [23], the issue has been considered analyzing both phase-space and distance-type measures of nonclassicality. In [22] the criterium to quantify the nonclassical depth of a state |Ψ is taken as the minimum of distance of this state and that of the nearest classical (coherent) state. A straightforward procedure using this concept leads to the operational formula for the nonclassical degree, dm (Ψ ) = 1 − πQmax (Ψ ), where Qmax (Ψ ) stands for maximum of Husimi Q-function. For our SPPS, using Eq. (13) for w = 1, we find the result shown in Fig. 8. Note that the nonclassical degree of this state is small (dm ∼ 0.1) for R ∼ 1 and tends to its maximum value (dm = 1) for R → ∞, when the SPPS tends to the SPS. The same limit (dm = 1) has been obtained for the Pegg–Barnett PS [24] and our PS introduced in [1]. The influence of phase ϕ upon the degree of nonclassicality is very small (not shown in figures). 5. Comments and conclusion We have studied preparation of SPS for stationary (Section 2.1) and running fields (Section 2.2), its nonclassical properties (Section 3) and nonclassical depth
Fig. 8. Plot of nonclassical depth y = dm (|Φ + (R, θ1 , θ2 )) versus R, for λ = 1 and ϕ = π/4.
(Section 4). The results of the SPS are obtained from those of SPPS in the limit of large values of R = |α| and r = |z| plus the prescription r = ln(R/λ), λ 1 (cf. Section 2). As discussed in [1], the limit R → ∞ furnishes an ideal PS since reaching this limit involves infinite energy. However, convenient approximations of the ideal case may be useful in certain circumstances, in the same way as plane waves become useful in many situations. We have shown that the SPPS (and SPS) exhibits various nonclassical properties: oscillations in the statistical distribution Pn± (Section 3.1)— which are connected with interference in the phase space—and collapse and revival effect of the atomic inversion W (T ) (Section 3.2). These two effects diminish when R grows and vanish if R → ∞, when each component of the superposition becomes a PS having these characteristics. We have also obtained the Wigner distribution and the Husimi Q-function (Section 3.3): the first showing in which way the interference terms emerge in the scenario (cf. Fig. 6), exhibiting both components of the superposition and the interference terms, a result not observed in [9]. Comparison of our Wigner functions with those in [9] is difficult due to the absence of corresponding parameters in these schemes. Concerning with the Husimi Q-function it allows us to calculate the measure of nonclassicality using the criterium discussed in [22] (Section 4). It was shown that our superposed state
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becomes more nonclassical when R grows, and most nonclassical (dm = 1) in the limit R → ∞, which coincides with the result found in [9] using the Pegg– Barnett SPS. For completeness, Section 3.4 studied the phase distribution of the SPS. In principle the present procedure can be pursued further, by replicating the experimental setup. By passing a sequence of atoms with controlled velocities throughout the cavity we obtain: a superposition having four PPS-components when using two atoms; eight PPS-components when using three atoms; etc. In this case the output field state obtained from the j th atom works as the input field state for the next one. This procedure was applied to create “high-generation Schrödinger-cat states” inside a cavity [25]. This extension is also valid for traveling waves by replicating the MZI and considering the output state of the j th MZI as input state of the next one. Of course, decoherence effects imply serious restriction to these replications, at least in the present status of technology. Here, we have restricted ourselves to the simple case of SPPS and SPS containing two components, as given in Eq. (6).
Acknowledgements We thank Professor N. Zagury for discussions (A.A.), CAPES (A.A.) and CNPq (P.B.M., A.T.A., B.B.), Brazilian agencies, for partially supporting this work.
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