Cavity-QED-based scheme for verification of the photon commutation relation

We propose an experimental scheme to prove the photon commutation relation, [a, a†]=1. The scheme exploits interaction between a single-mode cavity field and three two-level atoms, two of which are prepared in an entangled state. The success of the scheme is subject to observation of the atoms in predetermined states after the interaction. We show, in particular, that a reasonably high success probability can be obtained with our scheme by preparing the cavity field in a superposition of two Fock states and choosing the interaction times appropriately.


Introduction
Recent progress in quantum state engineering has made the experimental realization of singlephoton creation operation a † and annihilation operation a possible [1]. The field states generated by such operations are usually termed photon-added and photon-subtracted states, respectively. Photon-added coherent and thermal states were first considered by Agarwal and Tara [2,3], who showed that these states exhibit non-classical properties that are absent in the original coherent and thermal states. The photon creation operation a † on coherent and thermal states was performed experimentally by Zavatta et al [4]- [6]. The same group achieved sequences of photon creation and annihilation operations, namely photon-annihilation-thencreation operation a † a and photon-creation-then-annihilation operation aa † [7]. They showed, in particular, that operations a † a and aa † applied to the same field state produce states exhibiting different photon statistics, proving experimentally the non-commutativity of operators a and a † . This, however, does not prove the exact commutation relation [a, a † ] = 1. The exact commutation relation can only be proven if one constructs a scheme that performs a coherent superposition of the two operations a † a and aa † . One such scheme was recently proposed [8] and realized experimentally [9], in which a coherent superposition of the two operations is achieved in a single-photon interferometer setup employing beam splitters and a parametric amplifier.
The commutation relation [a, a † ] = 1 is the most fundamental relation that defines quantum optics, and it is thus of fundamental importance to prove it experimentally. In this paper, we propose a cavity-quantum electrodynamics (QED)-based scheme to prove the commutation relation [a, a † ] = 1. A theoretical analysis of cavity-QED-based schemes that perform individual photon creation and annihilation operations and their combinations, including the commutation operation, has been given recently [10]. A cavity-QED-based scheme to realize the photon-annihilation-then-creation operation a † a and the creation-thenannihilation operation aa † has been proposed [11,12], exploiting interaction between two atoms and a cavity field. The scheme we propose here exploits interaction between three atoms, two of which are entangled initially, and a cavity field. A coherent superposition of the two operations, a † a and aa † , results from the initial entanglement of the two atoms and post-selection of the states of the atoms that erases information on whether photon subtraction occurs before or after photon addition. Like the scheme proposed by Kim et al [8], our scheme is probabilistic as it relies on post-selection of the states of the atoms after their interaction with the cavity field. We pay particular attention to the issue of the success probability of the scheme.
In section 2, we briefly review the fundamentals of the interaction between a two-level atom and a monochromatic radiation field. The scheme we propose is described in section 3. It is shown that the final state of the cavity field after its interaction with the three atoms is approximately equal to the state that results when the commutation operation aa † − a † a is applied to the initial cavity state, if the interaction times are chosen sufficiently short. By showing that the final state of the cavity field is the same as the initial cavity state, the commutation relation can thus be proven. The success probability of the scheme, however, is generally very small. In section 4, we show that the success probability of the scheme can significantly be enhanced if consideration is restricted to the case when the cavity field is prepared initially in a superposition of two Fock states. In section 5, we describe the experimental feasibility of the proposed scheme. Finally, section 6 presents a summary.

Atom-field interaction
We briefly review the atom-cavity field interaction and introduce our notation. The time evolution of the atom-field system is determined by In (1), g is the atom-field coupling constant assumed to be real, σ − and σ + are atomic ladder operators defined as σ − = |g e|, with |g and |e denoting the lower and upper states of the atom, and the photon annihilation and creation operators a and a † are given by Note that we leave f n of (3) unspecified, because our purpose is to propose an experimental scheme that proves the commutation relation [a, a † ] = 1.
If the initial state of the atom-cavity system is given by |g n c n |n , corresponding to the situation where initially the atom is prepared in |g and the cavity field is in an arbitrary pure state n c n |n , (1) yields If the initial state is given by |e n c n |n , then (1) yields In (4) and (5), C g (t), S g (t), C e (t) and S e (t) are field operators defined as S e (t) = a † sin(gt Equations (4) and (5) can, respectively, be expressed, with the help of (3), as and

The scheme
The scheme we propose to prove the commutation relation consists of three two-level atoms and a single-mode cavity. The three atoms enter the cavity and interact with the cavity field one after another. We label them atoms 1, 2 and 3 in the order they enter the cavity. Atoms 1 and 3 are prepared initially in a singlet Bell state 1 √ 2 (|e 1 |g 3 − |g 1 |e 3 ), while atom 2 is prepared in its upper state |e 2 . The cavity field, assumed to be resonant with the atomic transition, can be in any arbitrary pure state, |ψ c (t = 0) = n c n |n . The experiment proceeds as atoms 1, 2 and 3, one after another, enter the cavity and interact with the cavity field for times t 1 , t 2 and t 3 , respectively. Measurement is made on the three atoms after the interactions are over, and the experiment is accepted only if the atoms are measured to be in the state |e 1 |g 2 |e 3 .
Straightforward algebra yields that the final state of the cavity field, contingent upon postselection of the three atoms in |e 1 |g 2 |e 3 , is given by where N {} signifies that the unnormalized state inside the curly bracket should be normalized. The physical interpretation of (9) is straightforward. The first term inside the curly bracket on the right-hand side results from a series of interactions with atom 1 prepared in |e 1 remaining in |e 1 [C e (t 1 )], atom 2 making a transition |e 2 → |g 2 [S e (t 2 )], and atom 3 prepared in |g 3 making a transition |g 3 → |e 3 [S g (t 3 )], whereas the second term results from a series of interactions with atom 1 prepared in |g 1 making a transition |g 1 → |e 1 [S g (t 1 )], atom 2 making a transition |e 2 → |g 2 [S e (t 2 )], and atom 3 prepared in |e 3 remaining in |e 3 [C e (t 3 )]. The − sign between the two terms arises from the phase difference of π in the initial singlet Bell state of atoms 1 and 3. Substituting (6a)-(6d) into (9) and using (3), we can express (9) as We desire this state to be equal to the state |ψ desired = N {(aa † − a † a) n c n |n }. The commutation relation can then be proven by showing experimentally that this state |ψ c (t 1 , t 2 , t 3 ) coincides with the initial cavity state n c n |n . It can be easily seen that, if one takes short interaction times such that wheren denotes the average number of photons in the initial cavity field, (10) becomes and our mission is accomplished. (Strictly speaking, the limit of short interaction times represented by (11) should be given in terms of n max , the maximum number of photons in the initial radiation field for which |c n | is non-negligible, instead of in terms ofn. We assume, however, that the radiation field has a smooth, continuous distribution of photon numbers, in which case the condition | f n max |gt 1 implies | fn|gt 1.) We can understand this result by noting that, in the limit of short interaction times, we have C g (t) 1, C e (t) 1, S g (t) (gt)a and S e (t) (gt)a † , and (9) becomes (12). Note that, due to the normalization in (12), the commutator [a, a † ] = aa † − a † a is realized by our scheme only up to a constant. By showing that the cavity field state after the interaction with the three atoms, with the interaction times t 1 , t 2 and t 3 chosen to be sufficiently short to satisfy (11), is identical to the initial cavity state, we can thus prove [a, a † ] = K , where K is an arbitrary constant. To complete the proof, one needs to show that the constant K is indeed one.
One way to prove K = 1 is to measure the success probability, i.e. the probability of observing the atoms in the desired atomic state |e 1 |g 2 |e 3 after the interaction. The success probability of our scheme can be easily calculated to be given by In the limit (11) of short interaction times, we obtain Noting that | f n+1 | 2 = n|aa † |n and | f n | 2 = n|a † a|n , (14) reduces to Measurement of the success probability thus determines the value of the constant K . As described above, in order to prove the commutation relation with our scheme, one needs to go to the limit (11) of short interaction times. Since the success probability P ∝ (gt) 2 (gt 2 ) 2 , our scheme suffers from a very low success probability. The problem of low success probability also exists in the optical scheme proposed by Kim et al [8]. This problem of low success probability may not be a serious threat to the optical scheme, because the traveling optical field can be provided at a high pulse rate. For the cavity-QED-based scheme, however, atoms are provided at a relatively low rate, and therefore it is desirable to find a way of enhancing the success probability. In the next section, we demonstrate that if the cavity field is prepared initially in a superposition of a small number (two) of Fock states, the success probability can be significantly enhanced.

The case when the cavity field is prepared in a superposition of two Fock states
In this section, we consider the simple case in which the cavity field is initially prepared in a superposition of two Fock states, α|N 1 + β|N 2 . Substituting c n = αδ n N 1 + βδ n N 2 (δ denotes the Kronecker delta function) into (10), we obtain In order for our scheme to realize the commutator [a, a † ], this state needs to be the same as the The success probability in this case is given, according to (13), by If (17) is satisfied, the success probability reduces to The problem reduces to finding the interaction times t 1 , t 2 and t 3 that satisfy (17). One obvious choice is the limit (11) of short interaction times. With such short interaction times, one sees that (17) is satisfied with Q (gt)(gt 2 ) and (19) , which is consistent with (14). As pointed out earlier, 7 however, this success probability is very low. It should be noted from (19) that P is of the order of Q 2 . It can be seen from (17) that if one can find non-small values of t 1 , t 2 and/or t 3 that satisfy (17), then Q and consequently P may not necessarily be small. In order to pursue this possibility, we consider the simplest case of N 1 = 0 and N 2 = 1. In this case, condition (17) becomes Let us take the limit (11) of short interaction times for t 1 and t 3 (| fn|gt 1 = | fn|gt 3 ≡ | fn|gt 1) but leave t 2 arbitrary. Equation (20) then becomes Although f 1 and f 2 are left unspecified, it can be seen that it is not too difficult to find nonsmall values of gt 2 that satisfy (21). Q is then of the order of (gt) and P is of the order of (gt) 2 . Although this success probability is still small, it represents a significant enhancement over P ∝ (gt) 2 (gt 2 ) 2 , the success probability as given by (15)  In summary, our scheme to prove the commutation relation [a, a † ] = K with the success probability significantly higher than the scheme of section 3 goes as follows. As in the scheme of section 3, we prepare atoms 1 and 3 in the entangled state 1 √ 2 (|e 1 |g 3 − |g 1 |e 3 ) and atom 2 in |e 2 . The cavity, however, is prepared in a superposition of two Fock states, for example, in a superposition of the vacuum and the one-photon state. We then let the three atoms interact with the cavity field one after another for interaction times t 1 , t 2 and t 3 , respectively. We post-select the case when the atoms after their interaction with the cavity field are observed to be in the state |e 1 |g 2 |e 3 . We choose the interaction times in such a way that (17) is satisfied. For the case when the cavity is prepared initially in a superposition of the vacuum and the one-photon state, we can choose gt 2 = 3.7613, while gt 1 = gt 3 ≡ gt is chosen sufficiently small to satisfy (11). The success probability of the scheme will then be of the order of (gt) 2 . Or we can choose gt 1 = gt 2 = gt 3 = 2.70 if an error of ∼ 15% can be tolerated. The success probability of the scheme will then be ∼ 0.014. By showing that the final cavity state after interaction with the atoms is identical to the initial cavity state, we prove experimentally that [a, a † ] = K .
To prove K = 1, we can exploit interaction between a single atom and a cavity field. We prepare the atom in |e and the cavity field in vacuum, |ψ(t = 0) = |e |0 . We let the atom 8 interact with the cavity field for time t and post-select the case when the atom is observed to remain in |e after the interaction. The probability of finding the atom in |e is given by Noting that | f 1 | 2 − | f 0 | 2 = 0|aa † − a † a|0 = K and that a|0 = 0 and therefore f 0 = 0, equation (22) can be written as One can therefore determine the value of K by monitoring the periodic variation of the success probability as the interaction time is varied and determining the period of the variation.

Experimental feasibility
In this section, we discuss the experimental feasibility of our scheme by giving realistic considerations to some experimental issues. In order to prove the commutation relation with the proposed scheme, it is necessary to verify that the final cavity state produced by the scheme is identical to the initial cavity state. Since the scheme operates in the strong coupling regime, direct measurement of the field state inside the cavity is not allowed. One thus needs to rely on probe atoms to obtain information on the field state. One such method that allows direct measurement of the Wigner distribution function of the cavity field is proposed [13] and demonstrated [14] experimentally for the vacuum and the one-photon state. The theory given in the previous sections assumes a lossless cavity and neglects atomic decay. In this regard, we note that our scheme needs to operate in the strong coupling regime in which the atom-field coupling constant (or the vacuum Rabi frequency) g is large compared with the cavity decay rate κ and the atomic decay rate γ . For the scheme described in section 3, the interaction time t needs to be chosen sufficiently small that gt 1. Since in the strong coupling regime g is large compared with κ and γ , the inequalities κt 1 and γ t 1 are well satisfied. For the scheme of section 4, the parameter gt may be of order unity, gt 1, but neglecting the cavity decay and atomic decay can still be justified. Let us take as an example the rubidium atom in a high-Q microwave cavity operating on hyperfine levels. We have g 10-100 kHz, whereas κ 0.01-1 kHz (corresponding to Q 3 × 10 10 -3 × 10 8 ) and γ 0.1-10 kHz [15]- [17], and thus the conditions κt 1 and γ t 1 can easily be satisfied. If we consider cesium atoms operating in the optical regime, we have, for example, g 34 MHz, κ 4.1 MHz and γ 2.5 MHz [18]. Although both cavity decay and atomic decay are more important in the optical regime, it is still possible to operate under the condition where they play relatively insignificant roles.
Perhaps the most demanding part of the scheme is the initial preparation of the atomic states and the cavity field states. Let us first consider the initial atomic states. Two atoms are required to be prepared in a maximally entangled singlet Bell state. We recall that it is exactly the entangled atoms in a singlet Bell state that were generated in the experiment in which generation of entanglement in massive particles was first demonstrated [19]. Subsequently, various elaborate cavity-QED methods of generating entangled atoms have been proposed and shown to be experimentally feasible [20]- [25]. Recently, entangled atoms or ions have been generated to teleport unknown atomic states [26,27]. One can therefore say that it is within the reach of current experimental technology to prepare atoms in the entangled state required by our scheme. There, however, remains a problem because, due to experimental errors, the 9 entangled state actually prepared may not exactly coincide with the desired entangled state. One possible source of error is the uncertainty in the interaction time arising from the finite resolution of the atomic velocity. Let us consider, for example, the scheme of [19] to prepare two atoms in the singlet Bell state. Two atoms, atoms A and B, prepared in the excited state and in the ground state, respectively, interact one after the other with a cavity prepared initially in vacuum. The interaction times, t A and t B , for atoms A and B, respectively, are chosen such that gt A = π 4 and gt B = π 2 . Due to experimental error, however, one may actually have gt A = π 4 (1 + A ) and gt B = π 2 (1 + B ), where A 1 and B 1. The state of the system, atom A + atom B + cavity field, after the interactions then becomes 1 Post-selection of the cavity field in vacuum results in the two atoms in the entangled state 1 Atoms A and B now play the role of atoms 1 and 3 in our proposed scheme. The initial state of the three atoms for our scheme is then given by 1 where ≡ A represents the error arising from a finite velocity resolution of the π 4 atomic pulse. Through straightforward algebra, we obtain that the final state of the cavity field produced by our scheme of section 3 is given, under the limit (11) of short interaction times, by The fidelity of this state with respect to the desired state |ψ desired = (aa † − a † a) n c n |n = n c n |n , defined as F = | ψ desired |ψ c (t 1 , t 2 , t 3 ) | 2 , is then given by F 1 − π 2 8 2 (2n + 1) 2 . This indicates that if our scheme is to work accurately against the error , the inequality 2 (2n + 1) 2 1, i.e.n 1/2 , must be satisfied. Taking 0.005 corresponding to a typical velocity resolution of 0.5% [28], we must haven 100. The experimental error in the initial preparation of atomic entanglement puts restrictions on the average photon number of the initial field state. We can also carry out a similar analysis for the scheme of section 4. Taking the initial atomic state to be given by (24) and taking the initial cavity field state as α|0 + β|1 , we obtain for the final cavity state, under the condition of short interaction times t 1 and t 3 , The fidelity of this state with respect to the desired state |ψ desired = α|0 + β|1 is given by The fidelity remains very close to 1 for 0.005. We can conclude that our scheme is not very sensitive to the error in the initial preparation of atomic entanglement, as long as the average photon number of the initial cavity state is not too large. In this respect, our scheme favors smalln.
As described in section 4, if our scheme is to perform with a reasonably high success probability, the cavity field needs to be prepared in a superposition of two Fock states. Various schemes to generate an arbitrary finite superposition of Fock states have been proposed [29]- [36] and demonstrated experimentally [37]- [39] in the past. In particular, an arbitrary superposition of the vacuum and the one-photon state can be generated using the optical quantum scissors device or other methods, as already demonstrated experimentally [37]- [39]. It appears therefore that the requirement to prepare the cavity field in a superposition of two Fock states is also within the reach of current experimental technology, although it is certainly not an easy requirement to be fulfilled.
Finally, we consider the effect due to the error in atomic state detection. Atomic detectors have miscounts of 2-5% [16,40]. This is potentially a serious problem for our scheme, because the miscount probability may be significantly higher than the success probability of the scheme and, as a consequence, it may be difficult to discern the success of the scheme from misfiring of the detector. We note, however, that the miscount rate can be made arbitrarily low at the expense of the detector efficiency and higher miscount probabilities of complementary detectors [40]. In order to quantitatively estimate the effect of the error in atomic state detection, we consider the case in which atoms 1 and 2 are post-selected correctly but atom 3 is post-selected incorrectly, i.e. the case when atom 3 in |g 3 is incorrectly identified as in |e 3 . The incorrect post-selection of atom 3 yields, for the final state of the cavity field, as opposed to (9), which results when the correct post-selection is made. Under the limit (11) of short interaction times, (27) reduces to |ψ c (t 1 , t 2 , t 3 ) N {a † n c n |n }. The fidelity of this state with respect to state (aa † − a † a) n c n |n = n c n |n is given by which reduces, for the initial coherent state, to F =n/(n + 1). One thus sees that ifn is large, the final cavity state produced by our scheme is insensitive to the error in atomic state detection. This insensitivity is rather undesirable, because erroneous data resulting from the detection error need to be discarded. The insensitivity makes it difficult to judge whether the coincidence of the final state produced by the scheme with the desired state actually proves the commutation relation or results simply from the insensitivity. One therefore sees that, in this respect, too, our scheme favors smalln. Ifn is large, it is difficult to detect the detection error simply by looking at photon statistics. One may have to rely on other more sensitive means, such as looking for the negativity of the Wigner distribution function. Another problem in atomic state detection in addition to misreading of atomic states is the possibility of atoms not being detected. In order to avoid errors arising from having more than one atom in a detection window, one needs to have atoms well separated in space from one another. When atoms are not detected, the experiment can be aborted. The detection efficiency of atomic detectors typically ranges from 50 to 90% [28], [40]- [44]. Taking, for example, the detection efficiency to be 0.8, the success probability of the scheme will be smaller by a factor of (0.8) 3 0.5 (the detection efficiency of one detector should be cubed because our scheme requires detection of three atoms) than the success probability with perfect detectors. The finite detection efficiency only lowers the success probability and it does not cause any error, as long as there is one or less than one atom in each detection window. Considering, however, that the proposed scheme generally has a low success probability, further lowering of the success probability is undesirable. It appears therefore that, for successful execution of the proposed scheme, it is important to minimize the error arising from misreading of atomic states and at the same time to maximize the detection efficiency of the atomic detectors.

Summary
We have proposed a cavity-QED-based scheme to prove the photon commutation relation [a, a † ] = 1. The scheme exploits interaction between a cavity field and three atoms, two of which are prepared in an entangled state. The proof is carried out in two steps. Firstly, the relation [a, a † ] = K , where K is an arbitrary constant, is proven by showing that the field state after the interaction, contingent upon post-selection of atoms in the desired state, is identical to the initial cavity state. Secondly, the constant K is proven to be one by measuring the success probability, i.e. the probability to find atoms in the desired state. If the scheme is to work for any arbitrary initial state of the cavity field, one needs to choose sufficiently short interaction times t 1 , t 2 and t 3 , as described in section 3. The choice of the short interaction times, however, leads inevitably to the problem of low success probability of the scheme. If we restrict our consideration to the case when the initial cavity state is prepared in a superposition of only two Fock states, it is then generally possible to choose non-small values of the interaction times that yield the final cavity state identical to the state that results when the commutation operation is applied to the initial cavity state, as described in section 4. The corresponding success probability is significantly higher than that of section 3. This higher success probability, however, comes at the expense of having to prepare the cavity field in a superposition of two Fock states.