- Email: [email protected]
Unambiguous discrimination of nonorthogonal entangled quantum states in cavity QED
Physics Letters A 383 (2019) 3069–3073
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Unambiguous discrimination of nonorthogonal entangled quantum states in cavity QED R.J. de Assis a , J.S. Sales b , N.G. de Almeida a,∗ a b
Instituto de Física, Universidade Federal de Goiás, 74.001-970, Goiânia - GO, Brazil Campus de Ciências Exatas e Tecnológicas, Universidade Estadual de Goiás, 75132-903, Anápolis, Goiás, Brazil
a r t i c l e
i n f o
Article history: Received 2 May 2019 Received in revised form 18 June 2019 Accepted 3 July 2019 Available online 10 July 2019 Communicated by M.G.A. Paris
a b s t r a c t In this work we propose a scheme in the context of superconducting cavities to unambiguously discriminate non-orthogonal quantum field states when one of them is one of the four Bell states. The present work, which generalizes an earlier proposal [1] dealing with non-entangled states, also makes use of a single high Q cavity, a sample of three-level atoms, Ramsey zones and selective detectors, thus consisting of an oversimplification from the experimental view. © 2019 Elsevier B.V. All rights reserved.
Keywords: POVM Generalized measurements Cavity QED Bell states
1. Introduction In a previous paper [1] we proposed an oversimplified scheme to unambiguously discriminate nonorthogonal quantum field states inside high-Q cavities relying on positive operator-valued measures (POVM). POVM, generalizing all possible kind of measurements [2,3], finds many applications in quantum computation [4,5], being particularly useful for quantum state discrimination [6–14] and quantum cryptography [15–18]. In this paper we extend the unambiguous non orthogonal quantum state discrimination technique in the context of cavity QED to encompass Bell entangled states, which are of paramount importance for quantum communication tasks, as for example superdense coding and teleportation [4]. Our scheme makes uses of a single bimodal cavity, three-level atoms [19–21] interacting with one mode at a time [22,23], Ramsey zones to manipulate the internal state of the three-level atoms, and selective atomic state detectors. Considering that all experiments in the cavity QED context, at the best of our knowledge, involved just one single (monomodal or bimodal) cavity, our proposal represents a considerable step toward the experimental feasibility of quantum state discrimination in this context. Different from previous schemes, our proposal makes no explicit use of POVM operators, making more comprehensible the theoretical analysis of quantum state discrimination. In this sense, our proposal is closely related to the so-called teleportation
*
Corresponding author. E-mail addresses: [email protected], nortonfi[email protected] (N.G. de Almeida).
https://doi.org/10.1016/j.physleta.2019.07.010 0375-9601/© 2019 Elsevier B.V. All rights reserved.
Fig. 1. Scheme of the experimental setup to unambiguously discrimination of nonorthogonal cavity-field states in cavity QED when one of the states is an entangled state. A three-level atom interacts with the two modes of a high Q bimodal cavity. The cavity mode fields are prepared in one of two nonorthogonal states |ψ1 and |ψ2 , with |ψ2 being a Bell state. Our protocol discriminates between these two nonorthogonal states prepared inside the bimodal high Q cavity.
schemes without the explicit use of Bell-State Measurements [24–27]. Note that although a POVM treatment is always possible, the approach pursued here aims to simplify a problem that otherwise would be near intractable. 2. Model We consider a three-level atom in ladder configuration, described by the set of states {|a , |b , |c }, and initially prepared in |b, see Fig. 1. The three-level atom enters a cavity interacting on-resonance with a singe mode of a cavity field which in turns is prepared either in state |ψ1 or in state |ψ2 , for which ψ1 |ψ2 = 0 (nonorthogonal states). Next, this three-level atom crosses two Ramsey zones (carrier interaction). Different from what
3070
R.J. de Assis et al. / Physics Letters A 383 (2019) 3069–3073
was done in Ref. [1], now we allow |ψ2 to be one the four of Bell states composing the so-called Bell basis ψ ± , φ ± , while the other state can be either |00 or |01. To be clear, our goal is then unambiguously to discriminate between (i) |ψ1 = |01 and |ψ2 = ψ ± = √1 (|01 ± |10); (ii) |ψ1 = |00 and |ψ2 = φ ± =
= 0. The joint atom-
and (3i) sin ( g 3 t ) ∓ cos ( g 3 t ) cos t 2g 12 + g 22 cavity field state then is found to be: (a) For initial states |ψ1 |b:
2
√1
2
(|00 ± |11), where either |ψ1 or |ψ2 are prepared in a bi-
modal high Q cavity and the three-level atom must interact with one mode at a time. Since the states |01 and φ ± as well as the ± states |00 and ψ are orthogonal, we will not consider them here. The one mode interaction at a time of the bimodal cavity is not an issue, and was addressed including experimentally in a number of papers, see for example [22,23,28,29]. After the atom passes through the cavity, it is detected in one of its three possible states, thus revealing in which state each mode of the cavity was prepared. The interaction with the two modes, one at a time, is controlled by the application of a Stark shift, which directs the atom to resonance with one mode while it is withdrawn from resonance with the other mode [22]. The dispersive interaction that necessarily appears when the atom is out of resonance with one mode will produce a small phase on the states that can be safely neglected [23,28]. The Hamiltonian model of the atom-field interaction into the cavity is given by H = H 1 + H 2 [30], where
H 1 = g1
σab a1 + σba a†1 + g2 σbc a1 + σcb a†1
(1)
|01 |b → −i cos ( g 3 t ) |a11 − sin ( g 3 t ) |b10; (b) For initial states |ψ2 |b:
± ψ |b → − √i
2
−
cos ( g 3 t ) ± sin ( g 3 t ) cos t
ig 1
2g 12
⎡
+
i ⎢ −√ ⎢ ⎣± 2
g 22
(3)
2g 12
+
|a11
g 22
sin t 2g 12 + g 22 |a20
g 2 sin t
2g 12
+
2g 12 + g 22
g 22
⎤ ⎥ ⎥ |c00 . ⎦
(4)
Note from (a) and (b) evolutions above that if the three-level atom is detected in |b the only consistent conclusion is that the cavity mode field was prepared in the state |ψ1 . Also, if the threelevel atom is detected in |c we have to conclude that the cavity mode field was prepared in state ψ ± ; on the other hand, since both |ψ1 and |ψ2 evolution leads to detection in |a, no conclusion can be drawn in this case. Note that this protocol can be used for |ψ2 being any one of ψ + and ψ − .
and
H 2 = g3
σab a2 + σba a†2 + g4 σbc a2 + σcb a†2 ,
†
operator and ai (ai ) is the creation (annihilation) photon number operator to the i-th mode. Our strategy consists in analyzing in which state the threelevel atom will be after crossing the cavity and being rotated in the Ramsey zones. Of course, this will depend on what state was prepared into the cavity, one of the two states we want to discriminate. After detecting the three-level atom, we proceed to a careful analysis of the parameters allowing us to associate to each state |ψi , i = 1, 2, a detection result |a, |b or |c . Exactly as occurs in POVM technique, there will be one detection related to the inconclusive result for quantum state discrimination and two results indicating unambiguous quantum state discrimination. The results, which depend on which pair of states were prepared within the cavity, are as follows:
(i) Assuming prepared states |ψ1 = |01 and |ψ2 = ψ ± = 2
√1
(2)
where gk , k = 1, 2, 3, 4, are the dipole couplings between the atom and fields - see Fig. 1, σην = |η ν | is the atom raising or lowering
√1
(|01 ± |10).
2
ters adjusted to (1ii) cos t 2g 12 + g 22
= 0, (2i) cos ( g 1 t ) = 0
= 0 or t 2g 12 + g 22 =
(2n2 + 1) π2 , n2 = 0, 1, ... (2ii) sin ( g 1 t ) = 0 or g 1 t = n1 π , n1 =
1, 2... (3ii) sin t 3g 32 + 2g 42 = 0, or t 3g 32 + 2g 42 = n4 π , n4 = 1, 2... (4ii) cos ( g 3 t ) = 0, or g 3 t = (2n3 + 1) π2 , n3 = 0, 1, ... and g γ (5ii) δ ∓ 2 = 0, where we are using the minus sign for 2g 2 + g 22
1 |ψ2 = ψ + = 2
To discriminate between these cavity field states, we assume the three-level atom enters the cavity in state |b, interacting first with mode 1 and then with mode 2. After that, the three-level atom is detected in one of its three states {|a , |b , |c }. Note the simplicity of this protocol: no Ramsey zone after the atom crosses the cavity is needed, just enough interaction with the two modes, one at a time, and selective detection. Therefore, for this case (i) involving entangled states, the discrimination protocol is even simpler than that in protocol Ref. [1]. After a straightforward calculation we will find the desired evolutions by adjusting parameters the following three con such that
(|00 ± |11).
Now, a slightly modification is in order: two Ramsey zones are placed after the cavity, such that the steps are then: the three-level atom enters the cavity in state |b, interacts with the cavity mode fields 1 and 2, crosses two Ramsey zones such that the atom state evolve in the first Ramsey zone as |a → γ1 |a + i δ1 |b and |b → i δ1 |a + γ1 |b with γ1 = 0 δ1 = 1; and in the second Ramsey zone the atom state evolves as |c → i δ |b + γ |c and |b → γ |b + i δ |c , with δ, γ ∈ R. Let us detail the steps until the atom is detected: the atom then crosses the cavity and the Ramsey
zones with parame-
√1
ditions are satisfied: (1i) sin t 2g 32 + g 42
(ii) Assuming prepared states |ψ1 = |00 and |ψ2 = φ ± =
√1
(|00 − |11).
2
(|00 + |11) and the plus sign for |ψ2 = ψ − =
Again, straightforward calculation now leads to (a) For initial states |ψ1 |b:
|00 |b → γ |b01 + i δ |c01 .
(5)
(b) For initial states |ψ2 |b
√ 2
± 2 δ φ |b → ∓ ig 1 |a20 + + γ |b01 . 2 γ 2g 12 + g 22
(6)
Now, note that from the evolutions indicated in (a) and (b) above, we can conclude that if the three-level atom is detected in |a the
R.J. de Assis et al. / Physics Letters A 383 (2019) 3069–3073
3071
Table 1 Values for the probabilities and parameters of the Hamiltonian model to according to our protocol to accomplish UQSD in cavity QED for states |ψ1 = |01 and |ψ2 = ψ ± . The table shows p u , p b , p c , and pd separately. The percentage of product (|01) and entangled states (ψ ± ) is equal: q = 0.5. Here we used g 1 t = π /2. q
p inc
pb
pc
pd
g2 t
g 3 t (+)
g 3 t (−)
g4 t
0.3 0.5 0.7
0.6558 0.7106 0.6500
0.0001 0.1417 0.3500
0.3411 0.1477 0.0000
0.3412 0.2894 0.3500
13.94 14.65 2.21
0.02 5.72 5.49
3.12 3.70 3.93
9.86 5.65 6.08
only consistent conclusion is that the cavity mode field was prepared in the entangled state |ψ2 , while if the three-level atom is detected in |c we conclude that the cavity mode field was prepared in state |ψ1 . Also, since both |ψ1 and |ψ2 evolution leads to detection in |b, no conclusion can be drawn in this case. 3. Discussion In the previous Section we have shown how to unambiguously discriminate between the nonorthogonal states |01 and ψ ± or ± |00 and φ . In this Section we show how to maximize probabilities of success by choosing the set of parameters properly. As we do not know in advance which state was prepared, we represent the cavity modes field state by ρ = q |ψ1 ψ1 | + (1 − q) |ψ2 ψ2 |, where q is the classical probability related to the frequency of preparing the state |ψ1 and (1 − q) for preparing |ψ2 . Next, let us analyze the cases (i) and (ii) of previous Section. (i) Discriminating between the nonorthogonal states |ψ1 = |01 and |ψ2 = ψ ± = √1 (|01 ± |10). From Eqs. (3) and (4), we see 2
that the probabilities are
p b = Tr (|b b| ρ ) = q sin2 ( g 3 t ) ,
(7)
of finding for sure that the state was |ψ1 = |01;
p c = Tr (|c c | ρ ) =
(1 − q )
g 22 sin2 t 2g 12 + g 22 (8)
2g 12 + g 22
2
of finding for sure that the state was |ψ2 = ψ ± . The inconclusive results can also be calculated from Eqs. (3) and (4), and easily checked to be pa = p inc = 1 − p b − p c . To optimize the protocol we can either minimize pa or maximize p b and p c . As there are too many variables, we choose to rewrite the success probabilities pa and p b in terms of g 2 t. Also, from condition (2i), g 1 t = (2n1 + 1) π2 , we write:
p c = (1 − q)
g 22 t 2 sin2
2 (2n1 + 1) π2 + g22 t 2
g 3 t = arctan ± cos we can write:
⎧ ⎨
(9)
(2n1 + 1)2 π 2 + 2g22 t 2
Now, from condition (3i), cos
2
2g 12 t 2
2g 12 t 2 + g 22 t 2
+
= tan ( g 3 t ), with
, and using g 1 t = (2n1 + 1) π2
⎞⎤⎫ ⎬ π p b = q sin2 arctan ⎣cos ⎝ (2n1 + 1)2 + g22 t 2 ⎠⎦ ⎩ ⎭ 2 ⎡
preparing state |ψ2 , we see that for g 2 t ∼ 2π (times usually used in experiments) this rate increases when the quantity of product states surpass that of entangled states. From g 2 t > 2π the success probability oscillates showing maxima for q = 0.7 (when the quantity of product states surpasses that of entangled states) coinciding with minima of q = 0.3 (when the quantity of entangled states surpasses that of product states). In Table 1 we display all the parameters required for our protocol to work. To build this table we take from the graph the value of g 2 t that maximizes the probability of success pd . Then we set the value of g 1 t by making n1 = 0 and with the value of g 2 t we use conditions (1ii) - (4ii) to obtain g 3 t. The probabilities p b and p c are obtained separately, such that pd = pb + p c . We now turn to the second case: (ii) Discriminating between the nonorthogonal states |ψ1 = |00 and |ψ2 = φ ± = √1 (|00 ± |11). From Eqs. (5) and (6), we 2
see that the probabilities now are
pa = Tr (|a a| ρ ) = q
g 12 t 2 2g 12 t 2 + g 22 t 2
⎛
2
(10)
We can now plot the success probability pd = p b + p c for discriminating both states |ψ1 and |ψ2 . In Fig. 2 we show pd versus g 2 t for q = 0.3, solid red line, q = 0.5, dotted blue line, and q = 0.7, dashed black line. Our numerical analysis show that the best probability occurs for n1 = 0. Since q is related to the probability of preparing state |ψ1 while 1 − q is related to the probability of
(11)
,
of finding for sure that the state was |ψ2 = φ ± and
p c = Tr (|c c | ρ ) = (1 − q)δ 2
g 22 t 2
Fig. 2. States discrimination success probability pd versus g 2 t for q1 = 0.3, solid red line, q = 0.5, dotted blue line, and 0.7, dashed black line. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
(12)
of finding for sure that the state was |ψ1 = |00. Again, inconclusive results can be calculated from Eqs. (5) and (6) or immediately from p b = p inc = 1 − pa − p c . Note, now, that unlike the prewe discriminate between states |ψ1 = |01 vious case in which and |ψ2 = ψ ± ), now the conditions imposed for the protocol to work, for g i t, i = 1, 2, 3, 4 are fixed. In fact, note that from condition (1ii)-(4ii) we fix g 1 t, which in turn, in condition (1i) fix g 2 t. Likewise, condition (4i) fixes g 3 t which in turn, used in con-
dition (3i), sets g 4 t. Also, from δ ∓ g 2 γ / 2g 12 + g 22 = 0 and noting
that δ 2 + γ 2 = 1 we can find γ and δ . As an example, in Table 2 we show the relevant parameters used to discriminate φ ± . As a final remark, we note that if we choose to project the cavity field state on the computational basis |i j , this would allow us to discriminate just one state. In fact, if the couple of state to be
3072
R.J. de Assis et al. / Physics Letters A 383 (2019) 3069–3073
Table 2 Values for probabilities and parameters of the Hamiltonian model according to our protocol to accomplish UQSD in cavity QED for |ψ1 = |00 and |ψ2 = φ ± . The percentage of product (|01) and entangled states (φ ± ) is equal: q = 0.5. The signal + (−) is for
+ − φ (φ ). Here we used n3 = 0 and n4 = 2 and g3 t = π , g4 t = 2 n1
n2
p inc
pc
pa
pd
g1 t
g2 t
γ
±δ
1 2 1
1 3 3
0.6589 0.7079 0.6566
0.0300 0.1288 0.3189
0.3111 0.1632 0.0245
0.3411 0.2920 0.3434
π 2π π
π
0.9487 0.8616 0.7378
0.3162 0.5075 0.6749
2
then whenever the projective measurement results |01 it is impossible do discern which state was prepared into the cavity. The same occurs to |ψ1 = |00 and |ψ2 = φ ± = √1 (|00 ± |11): 2
in this case no conclusion can be drawn if the projective measurement result |00. Also, note that by adopting this strategy, we would find a success probability of p s = Tr (|i j i j | ρ ) = 0.25, which is even lesser than the strategy we developed here - see Table 1 and Table 2. Besides, to the best of our knowledge, there is no technique allowing to directly project a cavity field state onto the computational basis, without ancillary atoms [28,31]. As a final remark, we note that strategy the strategy we presented here was build using well-known matter-radiation interaction parameters. It remains an open question if interactions appearing from effective Hamiltonians [29,32,33] could attain optimal results [14]. Before we conclude this Section, let us analyze the best rate of success one can obtain by using either projective measurements or POVM strategy. The state to be discriminated is ρ = q |ψ1 ψ1 | + (1 − q) |ψ2 ψ2 |. To compare with the well-known case of quantum state discrimination in two-dimensional Hilbert space [1,4], we call attention to the case where both nonorthogonal states |ψ1 = |00 and |ψ2 = √1 (|00 ± |11) are prepared 2
with equal probabilities, i.e. q = 1/2. The case when the states to be discriminated are |ψ1 = |01 and |ψ2 = √1 (|01 ± |10) is 2
similar and leads to the same conclusion. For projective measurements, since the only result enabling state discrimination is |11, the success probability rate is
= Tr |11 11| ρ =
1−q 2
(13)
.
P roj
For q = 1/2, p s = 1/4 and therefore it is the same as the similar case of considering a two-dimensional Hilbert space [1]. On the other hand, the POVM strategy can be pursued by building the POVM elements [1] for the enlarged Hilbert space:
E1 = E2 =
1 − |ψ1 ψ1 | 2
1 − |ψ2 ψ2 | 2
π to fulfill conditions (1ii)-(5ii) described in the main text.
0.3 0.5 0.7
P roj
5 2
q
discriminated is |ψ1 = |01 and |ψ2 = ψ ± = √1 (|01 ± |10),
ps
√
,
(14)
,
(15)
E 3 = 1 − E1 − E2 , (16) where the completeness relation reads 1 = + + k E k = 1 and ψ ψ + ψ − ψ − + φ + φ + + φ − φ − . Since Ei |ψi = 0, i = 1, 2, we conclude that whenever the measurement results E1 (E2 ) we can be certain that the state is |ψ2 (|ψ1 ). Thus, the success probability rate p s for the POVM strategy is p sP O V M = p 1 + p 2 , where p 1 = TrE1 ρ and p 2 = TrE2 ρ are the probabilities of occurrence to events E 1 and E 2 , respectively. √These probabilities are
easily seen to be equal to p 1 = p 2 =
q 2 √ , 2(1+ 2)
and therefore, for
q = 1/2 the POVM strategy leads to the optimal quantum state discrimination rate p sP O V M ∼ = 29.2%, which is greater than that of 25% for projective measurements. If we now compare with the values displayed in Table 1 and Table 2 for q = 1/2, we see that our scheme either achieves this optimal rate, Table 2, or is nearly optimal, Table 1.
2
2.1π 3.2π
4. Conclusion We have proposed a feasible scheme allowing to discriminate between two nonorthogonal cavity field states inside a bimodal high Q cavity, with one of them belonging to the set of states of the Bell base. Our protocol makes no explicit use of POVM operators, and allows us to achieve higher probabilities of success than projective measurements and nearly equal to the optimal POVM strategy. Our proposal relies on nowadays techniques in the cavity QED domain, making use of just one single three-level atom undergoing one cavity plus two Ramsey zones (carrier interaction) and one selective atomic state detector. This simplicity makes our protocol very attractive from the experimental point of view. Acknowledgements We acknowledge financial support from the Brazilian agency CNPq, CAPES and FAPEG. This work was performed as part of the Brazilian National Institute of Science and Technology (INCT) for Quantum Information. References [1] R.J. de Assis, J.S. Sales, N.G. de Almeida, Phys. Lett. A 381 (35) (18 September 2017) 2927–2933. [2] C.W. Helstrom, Quantum Detection and Estimation Theory, Academic, New York, 1976. [3] K. Jacobs, Quantum Measurement Theory and Its Applications, Cambridge University Press, 2014. [4] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000. [5] H.-K. Lo, S. Popescu, T.P. Spiller (Eds.), Introduction to Quantum Computation and Information, World Scientific Publishing, ISBN 981-02-3399-X, 1998. [6] J.A. Bergou, Quantum state discrimination and selected applications, J. Phys. Conf. Ser. 84 (2007) 012001. [7] I.D. Ivanovic, Phys. Lett. A 123 (6) (1987) 257–312. [8] A. Peres, Phys. Lett. A 128 (1) (1988) 1–2. [9] D. Dieks, Phys. Lett. A 126 (5–6) (11 January 1988) 303–306. [10] Asher Peres, Daniel Terno, J. Phys. A 31 (1998) 7105–7112. [11] A. Chefles, Quantum state discrimination, Contemp. Phys. 41 (2000) 401. [12] Anthony Chefles, Akira Kitagawa, Masahiro Takeoka, Masahide Sasaki, Jason Twamley, Unambiguous discrimination among oracle operators, J. Phys. A, Math. Theor. 40 (33) (2007) 10183–10213. [13] B. Huttner, A. Muller, J.D. Gautier, H. Zbinden, N. Gisin, Phys. Rev. A 54 (5) (1996). [14] G. Jaeger, A. Shimony, Phys. Lett. A 197 (1995) 83. [15] C.H. Bennett, G. Brassard, Quantum cryptography: public key distribution and coin tossing, Theor. Comput. Sci. 560 (Part 1) (4 December 2014) 7–11. [16] C.H. Bennett, G. Brassard, Quantum cryptography: public key distribution and coin tossing, in: Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, vol. 175, New York, 1984, p. 8. [17] S. Kak, A three-stage quantum cryptography protocol, Found. Phys. Lett. 19 (2006) 293–296. [18] A. Ekert, Phys. Rev. Lett. 67 (1991) 661–663. [19] Guang-Sheng Jin, Shu-Shen Li, Song-Lin Feng, Hou-Zhi Zheng, Generation of a supersinglet of three three-level atoms in cavity QED, Phys. Rev. A 71 (2005) 034307. [20] Dong-Sheng Ding, Zhi-Yuan Zhou, Bao-Sen Shi, Xu-Bo Zou, Guang-Can Guo, Generation of non-classical correlated photon pairs via a ladder-type atomic configuration: theory and experiment, Opt. Express 20 (10) (2012) 11433–11444. [21] Han Seb Moon, Taek Jeong, Three-photon electromagnetically induced absorption in a ladder-type atomic system, Phys. Rev. A 89 (2014) 033822. [22] A. Rauschenbeutel, et al., Phys. Rev. A 64 (2001) 050301.
R.J. de Assis et al. / Physics Letters A 383 (2019) 3069–3073
[23] Iara P. de Queirós, Wesley Bueno Cardoso, N.G. de Almeida, Superposition of coherent states prepared in one mode of a dissipative bimodal cavity, J. Phys. B, At. Mol. Opt. Phys. 40 (2007) 21–27. [24] L. Ye, G.-C. Guo, Scheme for teleportation of an unknown atomic state without the Bell-state measurement, Phys. Rev. A 70 (November 2004). [25] Shi-Biao Zheng, Scheme for approximate conditional teleportation of an unknown atomic state without the Bell-state measurement, Phys. Rev. A 69 (6) (June 2004). [26] W.B. Cardoso, Teleportation of GHZ-states in QED-cavities without the explicit Bell-state measurement, Int. J. Theor. Phys. 47 (4) (April 2008) 977–982. [27] R.W. Chhajlany, A. Wójcik, Comment on “Scheme for teleportation of an unknown atomic state without the Bell-state measurement”, Phys. Rev. A 73 (January 2006). [28] Iara P. de Queirós, Simone Souza, W.B. Cardoso, N.G. de Almeida, Phys. Rev. A 76 (2007) 0341011.
3073
[29] F.O. Prado, N.G. de Almeida, M.H.Y. Moussa, C.J. Villas-Boas, Bilinear and quadratic Hamiltonians in two-mode cavity quantum electrodynamics, Phys. Rev. A 73 (2006) 43803. [30] M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, 1997. [31] N.G. de Almeida, M.H.Y. Moussa, R.d.J. Napolitano, J. Phys. B, At. Mol. Opt. Phys. 44 (2011) 165502. [32] R.M. Serra, C.J. Villas-Boas, N.G. de Almeida, M.H.Y. Moussa, Frequency up- and down-conversion in two mode cavity quantum electrodynamics, Phys. Rev. A 71 (2005) 45802. [33] N.G. de Almeida, Engineering the unitary charge conjugation operator of quantum field theory for particle-antiparticle using trapped ions and light fields in cavity QED, J. Phys. B, At. Mol. Opt. Phys. 47 (2014) 165501.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Unambiguous discrimination of nonorthogonal entangled quantum states in cavity QED R.J. de Assis a , J.S. Sales b , N.G. de Almeida a,∗ a b
Instituto de Física, Universidade Federal de Goiás, 74.001-970, Goiânia - GO, Brazil Campus de Ciências Exatas e Tecnológicas, Universidade Estadual de Goiás, 75132-903, Anápolis, Goiás, Brazil
a r t i c l e
i n f o
Article history: Received 2 May 2019 Received in revised form 18 June 2019 Accepted 3 July 2019 Available online 10 July 2019 Communicated by M.G.A. Paris
a b s t r a c t In this work we propose a scheme in the context of superconducting cavities to unambiguously discriminate non-orthogonal quantum field states when one of them is one of the four Bell states. The present work, which generalizes an earlier proposal [1] dealing with non-entangled states, also makes use of a single high Q cavity, a sample of three-level atoms, Ramsey zones and selective detectors, thus consisting of an oversimplification from the experimental view. © 2019 Elsevier B.V. All rights reserved.
Keywords: POVM Generalized measurements Cavity QED Bell states
1. Introduction In a previous paper [1] we proposed an oversimplified scheme to unambiguously discriminate nonorthogonal quantum field states inside high-Q cavities relying on positive operator-valued measures (POVM). POVM, generalizing all possible kind of measurements [2,3], finds many applications in quantum computation [4,5], being particularly useful for quantum state discrimination [6–14] and quantum cryptography [15–18]. In this paper we extend the unambiguous non orthogonal quantum state discrimination technique in the context of cavity QED to encompass Bell entangled states, which are of paramount importance for quantum communication tasks, as for example superdense coding and teleportation [4]. Our scheme makes uses of a single bimodal cavity, three-level atoms [19–21] interacting with one mode at a time [22,23], Ramsey zones to manipulate the internal state of the three-level atoms, and selective atomic state detectors. Considering that all experiments in the cavity QED context, at the best of our knowledge, involved just one single (monomodal or bimodal) cavity, our proposal represents a considerable step toward the experimental feasibility of quantum state discrimination in this context. Different from previous schemes, our proposal makes no explicit use of POVM operators, making more comprehensible the theoretical analysis of quantum state discrimination. In this sense, our proposal is closely related to the so-called teleportation
*
Corresponding author. E-mail addresses: [email protected], nortonfi[email protected] (N.G. de Almeida).
https://doi.org/10.1016/j.physleta.2019.07.010 0375-9601/© 2019 Elsevier B.V. All rights reserved.
Fig. 1. Scheme of the experimental setup to unambiguously discrimination of nonorthogonal cavity-field states in cavity QED when one of the states is an entangled state. A three-level atom interacts with the two modes of a high Q bimodal cavity. The cavity mode fields are prepared in one of two nonorthogonal states |ψ1 and |ψ2 , with |ψ2 being a Bell state. Our protocol discriminates between these two nonorthogonal states prepared inside the bimodal high Q cavity.
schemes without the explicit use of Bell-State Measurements [24–27]. Note that although a POVM treatment is always possible, the approach pursued here aims to simplify a problem that otherwise would be near intractable. 2. Model We consider a three-level atom in ladder configuration, described by the set of states {|a , |b , |c }, and initially prepared in |b, see Fig. 1. The three-level atom enters a cavity interacting on-resonance with a singe mode of a cavity field which in turns is prepared either in state |ψ1 or in state |ψ2 , for which ψ1 |ψ2 = 0 (nonorthogonal states). Next, this three-level atom crosses two Ramsey zones (carrier interaction). Different from what
3070
R.J. de Assis et al. / Physics Letters A 383 (2019) 3069–3073
was done in Ref. [1], now we allow |ψ2 to be one the four of Bell states composing the so-called Bell basis ψ ± , φ ± , while the other state can be either |00 or |01. To be clear, our goal is then unambiguously to discriminate between (i) |ψ1 = |01 and |ψ2 = ψ ± = √1 (|01 ± |10); (ii) |ψ1 = |00 and |ψ2 = φ ± =
= 0. The joint atom-
and (3i) sin ( g 3 t ) ∓ cos ( g 3 t ) cos t 2g 12 + g 22 cavity field state then is found to be: (a) For initial states |ψ1 |b:
2
√1
2
(|00 ± |11), where either |ψ1 or |ψ2 are prepared in a bi-
modal high Q cavity and the three-level atom must interact with one mode at a time. Since the states |01 and φ ± as well as the ± states |00 and ψ are orthogonal, we will not consider them here. The one mode interaction at a time of the bimodal cavity is not an issue, and was addressed including experimentally in a number of papers, see for example [22,23,28,29]. After the atom passes through the cavity, it is detected in one of its three possible states, thus revealing in which state each mode of the cavity was prepared. The interaction with the two modes, one at a time, is controlled by the application of a Stark shift, which directs the atom to resonance with one mode while it is withdrawn from resonance with the other mode [22]. The dispersive interaction that necessarily appears when the atom is out of resonance with one mode will produce a small phase on the states that can be safely neglected [23,28]. The Hamiltonian model of the atom-field interaction into the cavity is given by H = H 1 + H 2 [30], where
H 1 = g1
σab a1 + σba a†1 + g2 σbc a1 + σcb a†1
(1)
|01 |b → −i cos ( g 3 t ) |a11 − sin ( g 3 t ) |b10; (b) For initial states |ψ2 |b:
± ψ |b → − √i
2
−
cos ( g 3 t ) ± sin ( g 3 t ) cos t
ig 1
2g 12
⎡
+
i ⎢ −√ ⎢ ⎣± 2
g 22
(3)
2g 12
+
|a11
g 22
sin t 2g 12 + g 22 |a20
g 2 sin t
2g 12
+
2g 12 + g 22
g 22
⎤ ⎥ ⎥ |c00 . ⎦
(4)
Note from (a) and (b) evolutions above that if the three-level atom is detected in |b the only consistent conclusion is that the cavity mode field was prepared in the state |ψ1 . Also, if the threelevel atom is detected in |c we have to conclude that the cavity mode field was prepared in state ψ ± ; on the other hand, since both |ψ1 and |ψ2 evolution leads to detection in |a, no conclusion can be drawn in this case. Note that this protocol can be used for |ψ2 being any one of ψ + and ψ − .
and
H 2 = g3
σab a2 + σba a†2 + g4 σbc a2 + σcb a†2 ,
†
operator and ai (ai ) is the creation (annihilation) photon number operator to the i-th mode. Our strategy consists in analyzing in which state the threelevel atom will be after crossing the cavity and being rotated in the Ramsey zones. Of course, this will depend on what state was prepared into the cavity, one of the two states we want to discriminate. After detecting the three-level atom, we proceed to a careful analysis of the parameters allowing us to associate to each state |ψi , i = 1, 2, a detection result |a, |b or |c . Exactly as occurs in POVM technique, there will be one detection related to the inconclusive result for quantum state discrimination and two results indicating unambiguous quantum state discrimination. The results, which depend on which pair of states were prepared within the cavity, are as follows:
(i) Assuming prepared states |ψ1 = |01 and |ψ2 = ψ ± = 2
√1
(2)
where gk , k = 1, 2, 3, 4, are the dipole couplings between the atom and fields - see Fig. 1, σην = |η ν | is the atom raising or lowering
√1
(|01 ± |10).
2
ters adjusted to (1ii) cos t 2g 12 + g 22
= 0, (2i) cos ( g 1 t ) = 0
= 0 or t 2g 12 + g 22 =
(2n2 + 1) π2 , n2 = 0, 1, ... (2ii) sin ( g 1 t ) = 0 or g 1 t = n1 π , n1 =
1, 2... (3ii) sin t 3g 32 + 2g 42 = 0, or t 3g 32 + 2g 42 = n4 π , n4 = 1, 2... (4ii) cos ( g 3 t ) = 0, or g 3 t = (2n3 + 1) π2 , n3 = 0, 1, ... and g γ (5ii) δ ∓ 2 = 0, where we are using the minus sign for 2g 2 + g 22
1 |ψ2 = ψ + = 2
To discriminate between these cavity field states, we assume the three-level atom enters the cavity in state |b, interacting first with mode 1 and then with mode 2. After that, the three-level atom is detected in one of its three states {|a , |b , |c }. Note the simplicity of this protocol: no Ramsey zone after the atom crosses the cavity is needed, just enough interaction with the two modes, one at a time, and selective detection. Therefore, for this case (i) involving entangled states, the discrimination protocol is even simpler than that in protocol Ref. [1]. After a straightforward calculation we will find the desired evolutions by adjusting parameters the following three con such that
(|00 ± |11).
Now, a slightly modification is in order: two Ramsey zones are placed after the cavity, such that the steps are then: the three-level atom enters the cavity in state |b, interacts with the cavity mode fields 1 and 2, crosses two Ramsey zones such that the atom state evolve in the first Ramsey zone as |a → γ1 |a + i δ1 |b and |b → i δ1 |a + γ1 |b with γ1 = 0 δ1 = 1; and in the second Ramsey zone the atom state evolves as |c → i δ |b + γ |c and |b → γ |b + i δ |c , with δ, γ ∈ R. Let us detail the steps until the atom is detected: the atom then crosses the cavity and the Ramsey
zones with parame-
√1
ditions are satisfied: (1i) sin t 2g 32 + g 42
(ii) Assuming prepared states |ψ1 = |00 and |ψ2 = φ ± =
√1
(|00 − |11).
2
(|00 + |11) and the plus sign for |ψ2 = ψ − =
Again, straightforward calculation now leads to (a) For initial states |ψ1 |b:
|00 |b → γ |b01 + i δ |c01 .
(5)
(b) For initial states |ψ2 |b
√ 2
± 2 δ φ |b → ∓ ig 1 |a20 + + γ |b01 . 2 γ 2g 12 + g 22
(6)
Now, note that from the evolutions indicated in (a) and (b) above, we can conclude that if the three-level atom is detected in |a the
R.J. de Assis et al. / Physics Letters A 383 (2019) 3069–3073
3071
Table 1 Values for the probabilities and parameters of the Hamiltonian model to according to our protocol to accomplish UQSD in cavity QED for states |ψ1 = |01 and |ψ2 = ψ ± . The table shows p u , p b , p c , and pd separately. The percentage of product (|01) and entangled states (ψ ± ) is equal: q = 0.5. Here we used g 1 t = π /2. q
p inc
pb
pc
pd
g2 t
g 3 t (+)
g 3 t (−)
g4 t
0.3 0.5 0.7
0.6558 0.7106 0.6500
0.0001 0.1417 0.3500
0.3411 0.1477 0.0000
0.3412 0.2894 0.3500
13.94 14.65 2.21
0.02 5.72 5.49
3.12 3.70 3.93
9.86 5.65 6.08
only consistent conclusion is that the cavity mode field was prepared in the entangled state |ψ2 , while if the three-level atom is detected in |c we conclude that the cavity mode field was prepared in state |ψ1 . Also, since both |ψ1 and |ψ2 evolution leads to detection in |b, no conclusion can be drawn in this case. 3. Discussion In the previous Section we have shown how to unambiguously discriminate between the nonorthogonal states |01 and ψ ± or ± |00 and φ . In this Section we show how to maximize probabilities of success by choosing the set of parameters properly. As we do not know in advance which state was prepared, we represent the cavity modes field state by ρ = q |ψ1 ψ1 | + (1 − q) |ψ2 ψ2 |, where q is the classical probability related to the frequency of preparing the state |ψ1 and (1 − q) for preparing |ψ2 . Next, let us analyze the cases (i) and (ii) of previous Section. (i) Discriminating between the nonorthogonal states |ψ1 = |01 and |ψ2 = ψ ± = √1 (|01 ± |10). From Eqs. (3) and (4), we see 2
that the probabilities are
p b = Tr (|b b| ρ ) = q sin2 ( g 3 t ) ,
(7)
of finding for sure that the state was |ψ1 = |01;
p c = Tr (|c c | ρ ) =
(1 − q )
g 22 sin2 t 2g 12 + g 22 (8)
2g 12 + g 22
2
of finding for sure that the state was |ψ2 = ψ ± . The inconclusive results can also be calculated from Eqs. (3) and (4), and easily checked to be pa = p inc = 1 − p b − p c . To optimize the protocol we can either minimize pa or maximize p b and p c . As there are too many variables, we choose to rewrite the success probabilities pa and p b in terms of g 2 t. Also, from condition (2i), g 1 t = (2n1 + 1) π2 , we write:
p c = (1 − q)
g 22 t 2 sin2
2 (2n1 + 1) π2 + g22 t 2
g 3 t = arctan ± cos we can write:
⎧ ⎨
(9)
(2n1 + 1)2 π 2 + 2g22 t 2
Now, from condition (3i), cos
2
2g 12 t 2
2g 12 t 2 + g 22 t 2
+
= tan ( g 3 t ), with
, and using g 1 t = (2n1 + 1) π2
⎞⎤⎫ ⎬ π p b = q sin2 arctan ⎣cos ⎝ (2n1 + 1)2 + g22 t 2 ⎠⎦ ⎩ ⎭ 2 ⎡
preparing state |ψ2 , we see that for g 2 t ∼ 2π (times usually used in experiments) this rate increases when the quantity of product states surpass that of entangled states. From g 2 t > 2π the success probability oscillates showing maxima for q = 0.7 (when the quantity of product states surpasses that of entangled states) coinciding with minima of q = 0.3 (when the quantity of entangled states surpasses that of product states). In Table 1 we display all the parameters required for our protocol to work. To build this table we take from the graph the value of g 2 t that maximizes the probability of success pd . Then we set the value of g 1 t by making n1 = 0 and with the value of g 2 t we use conditions (1ii) - (4ii) to obtain g 3 t. The probabilities p b and p c are obtained separately, such that pd = pb + p c . We now turn to the second case: (ii) Discriminating between the nonorthogonal states |ψ1 = |00 and |ψ2 = φ ± = √1 (|00 ± |11). From Eqs. (5) and (6), we 2
see that the probabilities now are
pa = Tr (|a a| ρ ) = q
g 12 t 2 2g 12 t 2 + g 22 t 2
⎛
2
(10)
We can now plot the success probability pd = p b + p c for discriminating both states |ψ1 and |ψ2 . In Fig. 2 we show pd versus g 2 t for q = 0.3, solid red line, q = 0.5, dotted blue line, and q = 0.7, dashed black line. Our numerical analysis show that the best probability occurs for n1 = 0. Since q is related to the probability of preparing state |ψ1 while 1 − q is related to the probability of
(11)
,
of finding for sure that the state was |ψ2 = φ ± and
p c = Tr (|c c | ρ ) = (1 − q)δ 2
g 22 t 2
Fig. 2. States discrimination success probability pd versus g 2 t for q1 = 0.3, solid red line, q = 0.5, dotted blue line, and 0.7, dashed black line. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
(12)
of finding for sure that the state was |ψ1 = |00. Again, inconclusive results can be calculated from Eqs. (5) and (6) or immediately from p b = p inc = 1 − pa − p c . Note, now, that unlike the prewe discriminate between states |ψ1 = |01 vious case in which and |ψ2 = ψ ± ), now the conditions imposed for the protocol to work, for g i t, i = 1, 2, 3, 4 are fixed. In fact, note that from condition (1ii)-(4ii) we fix g 1 t, which in turn, in condition (1i) fix g 2 t. Likewise, condition (4i) fixes g 3 t which in turn, used in con-
dition (3i), sets g 4 t. Also, from δ ∓ g 2 γ / 2g 12 + g 22 = 0 and noting
that δ 2 + γ 2 = 1 we can find γ and δ . As an example, in Table 2 we show the relevant parameters used to discriminate φ ± . As a final remark, we note that if we choose to project the cavity field state on the computational basis |i j , this would allow us to discriminate just one state. In fact, if the couple of state to be
3072
R.J. de Assis et al. / Physics Letters A 383 (2019) 3069–3073
Table 2 Values for probabilities and parameters of the Hamiltonian model according to our protocol to accomplish UQSD in cavity QED for |ψ1 = |00 and |ψ2 = φ ± . The percentage of product (|01) and entangled states (φ ± ) is equal: q = 0.5. The signal + (−) is for
+ − φ (φ ). Here we used n3 = 0 and n4 = 2 and g3 t = π , g4 t = 2 n1
n2
p inc
pc
pa
pd
g1 t
g2 t
γ
±δ
1 2 1
1 3 3
0.6589 0.7079 0.6566
0.0300 0.1288 0.3189
0.3111 0.1632 0.0245
0.3411 0.2920 0.3434
π 2π π
π
0.9487 0.8616 0.7378
0.3162 0.5075 0.6749
2
then whenever the projective measurement results |01 it is impossible do discern which state was prepared into the cavity. The same occurs to |ψ1 = |00 and |ψ2 = φ ± = √1 (|00 ± |11): 2
in this case no conclusion can be drawn if the projective measurement result |00. Also, note that by adopting this strategy, we would find a success probability of p s = Tr (|i j i j | ρ ) = 0.25, which is even lesser than the strategy we developed here - see Table 1 and Table 2. Besides, to the best of our knowledge, there is no technique allowing to directly project a cavity field state onto the computational basis, without ancillary atoms [28,31]. As a final remark, we note that strategy the strategy we presented here was build using well-known matter-radiation interaction parameters. It remains an open question if interactions appearing from effective Hamiltonians [29,32,33] could attain optimal results [14]. Before we conclude this Section, let us analyze the best rate of success one can obtain by using either projective measurements or POVM strategy. The state to be discriminated is ρ = q |ψ1 ψ1 | + (1 − q) |ψ2 ψ2 |. To compare with the well-known case of quantum state discrimination in two-dimensional Hilbert space [1,4], we call attention to the case where both nonorthogonal states |ψ1 = |00 and |ψ2 = √1 (|00 ± |11) are prepared 2
with equal probabilities, i.e. q = 1/2. The case when the states to be discriminated are |ψ1 = |01 and |ψ2 = √1 (|01 ± |10) is 2
similar and leads to the same conclusion. For projective measurements, since the only result enabling state discrimination is |11, the success probability rate is
= Tr |11 11| ρ =
1−q 2
(13)
.
P roj
For q = 1/2, p s = 1/4 and therefore it is the same as the similar case of considering a two-dimensional Hilbert space [1]. On the other hand, the POVM strategy can be pursued by building the POVM elements [1] for the enlarged Hilbert space:
E1 = E2 =
1 − |ψ1 ψ1 | 2
1 − |ψ2 ψ2 | 2
π to fulfill conditions (1ii)-(5ii) described in the main text.
0.3 0.5 0.7
P roj
5 2
q
discriminated is |ψ1 = |01 and |ψ2 = ψ ± = √1 (|01 ± |10),
ps
√
,
(14)
,
(15)
E 3 = 1 − E1 − E2 , (16) where the completeness relation reads 1 = + + k E k = 1 and ψ ψ + ψ − ψ − + φ + φ + + φ − φ − . Since Ei |ψi = 0, i = 1, 2, we conclude that whenever the measurement results E1 (E2 ) we can be certain that the state is |ψ2 (|ψ1 ). Thus, the success probability rate p s for the POVM strategy is p sP O V M = p 1 + p 2 , where p 1 = TrE1 ρ and p 2 = TrE2 ρ are the probabilities of occurrence to events E 1 and E 2 , respectively. √These probabilities are
easily seen to be equal to p 1 = p 2 =
q 2 √ , 2(1+ 2)
and therefore, for
q = 1/2 the POVM strategy leads to the optimal quantum state discrimination rate p sP O V M ∼ = 29.2%, which is greater than that of 25% for projective measurements. If we now compare with the values displayed in Table 1 and Table 2 for q = 1/2, we see that our scheme either achieves this optimal rate, Table 2, or is nearly optimal, Table 1.
2
2.1π 3.2π
4. Conclusion We have proposed a feasible scheme allowing to discriminate between two nonorthogonal cavity field states inside a bimodal high Q cavity, with one of them belonging to the set of states of the Bell base. Our protocol makes no explicit use of POVM operators, and allows us to achieve higher probabilities of success than projective measurements and nearly equal to the optimal POVM strategy. Our proposal relies on nowadays techniques in the cavity QED domain, making use of just one single three-level atom undergoing one cavity plus two Ramsey zones (carrier interaction) and one selective atomic state detector. This simplicity makes our protocol very attractive from the experimental point of view. Acknowledgements We acknowledge financial support from the Brazilian agency CNPq, CAPES and FAPEG. This work was performed as part of the Brazilian National Institute of Science and Technology (INCT) for Quantum Information. References [1] R.J. de Assis, J.S. Sales, N.G. de Almeida, Phys. Lett. A 381 (35) (18 September 2017) 2927–2933. [2] C.W. Helstrom, Quantum Detection and Estimation Theory, Academic, New York, 1976. [3] K. Jacobs, Quantum Measurement Theory and Its Applications, Cambridge University Press, 2014. [4] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000. [5] H.-K. Lo, S. Popescu, T.P. Spiller (Eds.), Introduction to Quantum Computation and Information, World Scientific Publishing, ISBN 981-02-3399-X, 1998. [6] J.A. Bergou, Quantum state discrimination and selected applications, J. Phys. Conf. Ser. 84 (2007) 012001. [7] I.D. Ivanovic, Phys. Lett. A 123 (6) (1987) 257–312. [8] A. Peres, Phys. Lett. A 128 (1) (1988) 1–2. [9] D. Dieks, Phys. Lett. A 126 (5–6) (11 January 1988) 303–306. [10] Asher Peres, Daniel Terno, J. Phys. A 31 (1998) 7105–7112. [11] A. Chefles, Quantum state discrimination, Contemp. Phys. 41 (2000) 401. [12] Anthony Chefles, Akira Kitagawa, Masahiro Takeoka, Masahide Sasaki, Jason Twamley, Unambiguous discrimination among oracle operators, J. Phys. A, Math. Theor. 40 (33) (2007) 10183–10213. [13] B. Huttner, A. Muller, J.D. Gautier, H. Zbinden, N. Gisin, Phys. Rev. A 54 (5) (1996). [14] G. Jaeger, A. Shimony, Phys. Lett. A 197 (1995) 83. [15] C.H. Bennett, G. Brassard, Quantum cryptography: public key distribution and coin tossing, Theor. Comput. Sci. 560 (Part 1) (4 December 2014) 7–11. [16] C.H. Bennett, G. Brassard, Quantum cryptography: public key distribution and coin tossing, in: Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, vol. 175, New York, 1984, p. 8. [17] S. Kak, A three-stage quantum cryptography protocol, Found. Phys. Lett. 19 (2006) 293–296. [18] A. Ekert, Phys. Rev. Lett. 67 (1991) 661–663. [19] Guang-Sheng Jin, Shu-Shen Li, Song-Lin Feng, Hou-Zhi Zheng, Generation of a supersinglet of three three-level atoms in cavity QED, Phys. Rev. A 71 (2005) 034307. [20] Dong-Sheng Ding, Zhi-Yuan Zhou, Bao-Sen Shi, Xu-Bo Zou, Guang-Can Guo, Generation of non-classical correlated photon pairs via a ladder-type atomic configuration: theory and experiment, Opt. Express 20 (10) (2012) 11433–11444. [21] Han Seb Moon, Taek Jeong, Three-photon electromagnetically induced absorption in a ladder-type atomic system, Phys. Rev. A 89 (2014) 033822. [22] A. Rauschenbeutel, et al., Phys. Rev. A 64 (2001) 050301.
R.J. de Assis et al. / Physics Letters A 383 (2019) 3069–3073
[23] Iara P. de Queirós, Wesley Bueno Cardoso, N.G. de Almeida, Superposition of coherent states prepared in one mode of a dissipative bimodal cavity, J. Phys. B, At. Mol. Opt. Phys. 40 (2007) 21–27. [24] L. Ye, G.-C. Guo, Scheme for teleportation of an unknown atomic state without the Bell-state measurement, Phys. Rev. A 70 (November 2004). [25] Shi-Biao Zheng, Scheme for approximate conditional teleportation of an unknown atomic state without the Bell-state measurement, Phys. Rev. A 69 (6) (June 2004). [26] W.B. Cardoso, Teleportation of GHZ-states in QED-cavities without the explicit Bell-state measurement, Int. J. Theor. Phys. 47 (4) (April 2008) 977–982. [27] R.W. Chhajlany, A. Wójcik, Comment on “Scheme for teleportation of an unknown atomic state without the Bell-state measurement”, Phys. Rev. A 73 (January 2006). [28] Iara P. de Queirós, Simone Souza, W.B. Cardoso, N.G. de Almeida, Phys. Rev. A 76 (2007) 0341011.
3073
[29] F.O. Prado, N.G. de Almeida, M.H.Y. Moussa, C.J. Villas-Boas, Bilinear and quadratic Hamiltonians in two-mode cavity quantum electrodynamics, Phys. Rev. A 73 (2006) 43803. [30] M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, 1997. [31] N.G. de Almeida, M.H.Y. Moussa, R.d.J. Napolitano, J. Phys. B, At. Mol. Opt. Phys. 44 (2011) 165502. [32] R.M. Serra, C.J. Villas-Boas, N.G. de Almeida, M.H.Y. Moussa, Frequency up- and down-conversion in two mode cavity quantum electrodynamics, Phys. Rev. A 71 (2005) 45802. [33] N.G. de Almeida, Engineering the unitary charge conjugation operator of quantum field theory for particle-antiparticle using trapped ions and light fields in cavity QED, J. Phys. B, At. Mol. Opt. Phys. 47 (2014) 165501.