Past of a quantum particle and weak measurement
aa r X i v : . [ qu a n t - ph ] J un Past of a quantum particle and weak measurement
Zheng-Hong Li,
1, 2, 3
M. Al-Amri,
1, 2 and M. Suhail Zubairy
1, 3 Institute for Quantum Science and Engineering (IQSE) and Department of Physics and Astronomy,Texas A&M University, College Station, Texas 77843-4242 The National Center for Mathematics and Physics,KACST, P.O.Box 6086, Riyadh 11442, Saudi Arabia Beijing Computational Science Research Center, Beijing 100084, China (Dated: October 10, 2018)We present an analysis of a nested Mach-Zehnder interferometer in which an ensemble of identicalpre- and postselected particles leave a weak trace. A knowledge of the weak value partially destroysthe quantum interference. The results, contrary to some recent claims, are in accordance with theusual quantum mechanical expectations.
I. INTRODUCTION
Weak measurement [1, 2], as its name implies, is akind of quantum measurement where the coupling be-tween the measured system and the measuring device isso weak that the system remains unaffected during theprocess of measurement. A single measurement does notprovide any information about the system but, after alarge number of repeated measurements on an ensembleof identically prepared pre- and postselected systems, in-formation can be extracted. The notions of weak mea-surement and weak values were first introduced in a clas-sic paper by Aharonov, Albert and Vaidman in 1988 [1].Since then this idea has found a number of interestingapplications in quantum measurement [3–7].The weak measurements and the physical meaning ofweak values remain a subject of arguments and discus-sion [8–15]. An example is shown in Fig. 1 where asmall Mach-Zehnder interferometer is inserted into onearm of a big interferometer. Let us assume that, if asingle photon is sent into the small interferometer at po-sition F, the detector D clicks with unit probability asa result of interference. In this case, if a photon is sentat the input and the detector D clicks, it is reasonableto assume that the single photon must have followed theouter path A and the probability of its existence insidethe smaller Mach-Zehnder interferometer (along paths F,B, C, and E) must be zero. This observation lies at therecent schemes for counterfactual computation [16] andcommunication [17]. However, it is argued in some recentpapers [18, 19] that this conclusion may not be correct.Using the concept of weak measurement it is shown that,in the case the detector D clicks, the probability of find-ing the photon is zero at locations F and E but is non-zeroinside the small interferometer along paths B and C. Inthe words of Vaidman[18], “The photon did not enter theinterferometer, the photon never left the interferometer,but it was there”. This is, to put it mildly, very surpris-ing and this surprise is compounded by the claim thatthe photon number in one arm of the small interferom-eter is 1 whereas in the other arm its value is -1. As aresult, the probability of finding the photon at positionE is zero. The objective of this paper is to resolve this mystery.We show that, although the method of weak measure-ment provides a very different angle to view quantumsystems, both its mathematics and internal essence obeythe usual approach to quantum mechanics. For the samequantum system, without the post-selected state, quan-tum mechanics provides us the probabilities associatedwith observable quantities. We show that the same istrue even when a weak measurement is made in a post-selected system.In the following, we will first present some general ar-guments to understand the consequences of the weakmeasurement on the system. We will then consider anexample of system-meter interaction that can possiblybe implemented via a dispersive atom-field interaction.Our analysis shows that the disturbance caused by the“weak” measurement partially destroys the interferenceand the probability of finding photon at E is not zeroanymore. This disturbance caused by a weak measure-ment is non-zero no matter how weak the measurementis. We also show that all the results can be understoodwithin the framework of a conventional quantum mechan-ical approach and the weak measurement does not addanything further in our understanding and interpretationof the system considered in Fig. 1. II. WEAK MEASUREMENT IN A DOUBLEMACH-ZEHNDER INTERFEROMETER
First we present a brief discussion about the weak mea-surement. Suppose there is a pointer P that is coupled toan observable A of a system via a Hamiltonian H = ¯ hη APwith a very weak coupling η . The Hamiltonian per-turbs the system state before measurement (pre-selectedstate | ψ i i ) as e − iHτ/ ¯ h | ψ i i ≈ | ψ i i − ( iHτ / ¯ h ) | ψ i i = | ψ i i − iηt A | ψ i i P. To extract the perturbation, we canproject it on a post-selected state | ψ f i , which is indepen-dent of the measurement and allows us to investigate thesystem with the special final state. This can also be un-derstand as a precondition. As a result, the weak valueis defined by A w = h ψ f | A | ψ i i / h ψ f | ψ i i . We note thatthe weak value is not a directly observable quantity in BS1
A B CF EL1L2 L3 D3 L4 > |1 > |0 > |0 BS2 BS2
D2D1
BS1(r,t) (r,t)(50,50) (50,50)
FIG. 1. (Color online) A small Mach-Zehnder interferometeris added in the upper arm of the big interferometer. A singlephoton pulse is sent into the setup. BS stands for beam-splitter. BS ’s reflectivity is r , transmissivity is t while BS is50%-50% beam-splitter. D , D and D stand for detectors.L1, L2, L3 and L4 stand for stages corresponding to timeevolution of the photon. any real experiment and is to be inferred from the dataof an actual experiment.To show the concept more clearly, we again turn to theexample as shown in Fig. 1. We consider the state of thephoton at four stages as shown by the dotted lines. Asingle photon is sent into the left side of the beam-splitter BS whose function can be described as U L : ( | i → r | i + t | i| i → r | i − t | i (1)where r and t are the reflectivity and transmissivity of BS and the photon number state | n , n , n i describesthe number of photons in the path at the left side of BS ,the path between two BS s and two BS s (including po-sition F, B, E) and the path at the right side of BS . For BS s, the transformation property is the same as BS ’sexcept that their reflectivity and transmissivity are equalto 1 / √ U L | i = r | i + t | i . Let the operators U L , U L , and U L describe operations between the two adjacent stages. Wealso assume that a weak measurement of the operator | i h | is made at position C. The pre-selected stateat stage L2 is | ψ i i = U L U L | i = r | i + t √ | i + | i ) , (2) and, under the condition that D clicks, the post-selectedstate is h ψ f | = h | U L U L = r h | − t √ h | − h | ) . (3)The weak value corresponding to a weak measurementat position C is A C = h ψ f | | i h | | ψ i ih ψ f | | ψ i i = t r (4)Similarly, we obtain the weak value corresponding tothe weak measurement at position B, which is A B = − t / (2 r ). A similar calculation also yields, for the mea-surements at positions A, E, and F, the weak values A E = A F = 0 and A A = 1. These results appear strangeat first glance as it seems that the photon appears at Band C but not at E.Before further discussion, we answer the importantquestion: What is the meaning of the weak value? Wenote that the denominator in the expression for weakvalue is h ψ f | | ψ i i = h | U L U L U L U L | i . It isclear that whether we calculate it as h ψ f | × | ψ i i or h | × ( U L U L U L U L | i ) (i.e., we let the systemevolve stage by stage until stage L4, and then projectsit on D ), we get the same result. The modulus squareof h ψ f | | ψ i i means the probability D clicking, which isequal to r . This implies that the photon passes a clas-sical path without interference. One may argue that theabove discussion is meaningless since the measurement isnot included. However, in the following we will show thateven if we include the measurement process, the conceptof post-selected state is still not necessary and the weakvalue actually tells us the level that the original systemis perturbed.The numerator in the expression forthe weak value is h ψ f | | i h | | ψ i i = h | U L U L | i h | U L U L | i . Thisexpression is no different if we calculate U L U L | i h | U L U L | i at first, and thenproject it on D . Nevertheless, the physical meaningof this quantity is clear: The photon state evolvesstage by stage until, at stage L2, a projection mea-surement is made so that the photon state collapses to | i . After that, the system evolves again but witha new initial state. The interference is destroyed as aresult of the projection measurement. The quantity | h ψ f | | i h | | ψ i i | represent the probability of D clicking under the condition that the photon is found atthe position C. In this case a click at D correspondsto a different situation in comparison with when we didnot try to obtain which-path information. Obviously,since we tried to obtain the which-path information, theinterference is lost.In a weak measurement, the evolution is givenby e − iHt | ψ i i ≈ | ψ i i − ( iHt ) / ¯ h | ψ i i = | ψ i i − iηt | i h | P | ψ i i , and the measurement does not dis-turb the original system too much. However, the weakvalue itself is not “weak”, which comes from the partialsystem whose interference is destroyed. In other words,the weak value leads to noise that stems from the mea-surement. In particular, the weak value can not tell usany information about the photon path without affectingnext evolution processes of the photon even if the finalstate does not change.Now we can roughly answer why A B , A C are non-zerobut A E = 0. The main reason is that these three mea-surements are not made on the same system at the sametime. If we do not make any measurement at B orC, there is no doubt no photon will be found at E. Astraightforward calculation without post-selected stateleads to the same result. However, if we make a mea-surement at B or C, the situation is different. Since in-terference is destroyed, the photon has some probabilityleaking into E and finally causes D to click. The prob-ability of finding the photon at E is not zero anymore.So far we have qualitatively discussed the effect of weakmeasurement and shown that the approach of weak mea-surement should not lead to a paradox that does not ex-ist in the usual quantum approach. In particular we havediscussed that the weak value describes the noise corre-sponding to measurement. In the following, we considera model to show that the conclusion that the photon ex-ists in paths B and C but not E is not correct and theexplanation in terms of negative photon numbers is notneeded. III. WEAK NON-DEMOLITIONMEASUREMENT
We consider a weak quantum non-demolition mea-surement using a Hamiltonian of the form H =¯ hηa c † a c | b i h b | , where the system operator is a † c a c = | i h | indicating the measurement is at position Cand the state | b i is the state of the meter.Such a Hamiltonian can, for example, be realized bya single three-level atom in the cascade configuration inthe arm C [20, 21]. The upper two-levels | a i and | b i are dispersively coupled to the photon with a detuning∆ such that η = ¯ hg / ∆ [22] with g being the atom-field coupling coefficient. The atom, acting as a meter, isinitially prepared in a superposition of the middle level | b i and the lower level | c i , i. e., | ψ A i = ( | b i + | c i ) / √ | α i | ψ f i (with D clicking and the atom found in level α with α = b, c ) is h α | h ψ f | e − iHτ/ ¯ h | ψ i i | ψ A i≈ h α | h ψ f | (1 − iHτ / ¯ h ) | ψ i i | ψ A i = h α | h ψ f | | ψ i i (cid:18) | b i + | c i√ − iA C ητ | b i√ (cid:19) ≈ h α | h ψ f | | ψ i i ( e − iηA C τ | b i + | c i ) / √ | b i → ( | b i + i | c i ) / √ | c i → ( i | b i + | c i ) / √ h i | h ψ f | | ψ i i (cid:2) e iηA C τ ( | b i + i | c i ) + ( i | b i + | c i ) (cid:3) /
2. Theprobability of finding the atom in the levels | b i and | c i are P b = 12 | h ψ f | | ψ i i | [1 − sin( ηA C τ )] , (6) P c = 12 | h ψ f | | ψ i i | [1 + sin( ηA C τ )] . (7)The weak value A C can now be inferred from P b and P c as follows: A C = 1 ητ arcsin P c − P b P b + P c . (8)We also note that | h ψ f | | ψ i i | = P b + P c , (9)Thus the weak value A C is obtained indirectly under thefirst order approximation. The probability of D clickingdoes not change as a result of weak measurement since | h ψ f | | ψ i i | = P b + P c . This makes it tempting to claimthat the weak measurement has no influence on the finaloutcome and we should be able to conclude that the pho-ton exists in path C but not E. The situation is howevermore complex and we look at it more carefully.The evolved state at stage L2 after the measurementis (1 − iHτ / ¯ h ) | ψ i i | ψ A i = (1 + i ) (cid:20) r | i + t √ | i + | i ) (cid:21) ( | b i + | c i ) / − i t √ ητ | i ( | b i + i | c i ) / ητ << | b i and | c i can notgive us significant information of the photon path. How-ever, the distribution of the photon in different paths haschanged corresponding to the measurement. The quan-tum interference is partially destroyed. For example, ifthe atom is found in level | b i , the probability of findingthe photon at E is η τ t /
16. Since the component ofthe photon state at position E corresponding to | b i and | c i have different phases, they have different contributionfor the interference happening at BS . In one case theprobability of D clicking increases, in the other case itdecreases resulting in a null effect. However, in the fol-lowing, we show that, if we carry out our calculation ex-actly (instead of restricting only to the first order in ητ ),the measurement changes the probability of D clickingin the order ( ητ ) .The photon-atom state at stage L2 is given by e − iHτ | ψ i i | ψ A i = ( r | i + t √ | i + | i )) | b i + | c i√ t (cid:0) e − iητ − (cid:1) | i | b i→ i r | i + t √ | i + | i ))( | c i + | b i )+ t √ (cid:0) e − iητ − (cid:1) | i ( | b i + i | c i ) . (11)in the last step, we made the same unitary transformationfor | b i and | c i as discussed above (namely | b i → ( | b i + i | c i ) / √ | c i → ( i | b i + | c i ) / √ U L U L e − iHτ | ψ i i | ψ A i = U L [ 1 + i r | i + t | i )( | c i + | b i )+ t (cid:0) e − iητ − (cid:1) ( | i − | i )( | b i + i | c i )]= 1 + i r | i + rt | i + t | i )( | c i + | b i )+ t (cid:0) e − iητ − (cid:1) ( | i − r | i + t | i )( | b i + i | c i ) . (12)Regardless of whether we find the atom in level | b i or | c i , the probability of finding the photon at E is t [1 − cos( ητ )]. The probability D clicking is nowgiven by P b + P c = r + t [1 − cos( ητ )] (cid:16) t − r (cid:17) ,where P b = (cid:12)(cid:12)(cid:12) i r + t (cid:0) e − iητ − (cid:1)(cid:12)(cid:12)(cid:12) and P c = (cid:12)(cid:12)(cid:12) i r + i t (cid:0) e − iητ − (cid:1)(cid:12)(cid:12)(cid:12) . An interesting observation isthat, if we project Eq. (11) directly on the post-selectedstate, we get the same result.Here we note that the probability of D clicking ischanged due to the influence of measurements. Inthe weak measurement approximation, these resultsreduce to P b ≈ r [1 − sin( A C ητ )] / P C ≈ r [1 + sin( A C ητ )] / D which comes from measurement dis-appears.So here is the resolution of the confusion. First wenote that, although the probability of finding the photonat E is second order in ητ , the amplitude is still first or-der. This linear amplitude proportional to the transmis-sivity ( − iητ t / r (1 + i ) / P b that is linear in ητ . Asimilar result is obtained for P c . This important obser-vation clarifies that, no matter how weak the quantumnon-demolition measurement at B or C is, it destroys theinterference leading to a non-zero amplitude at E, whichwhen combined with the amplitude along A gives the cor-rect detection probability at detector D . As P E appears to be zero in the linear approximation, we can be led tothe erroneous claim that the photon does not exist at Ewhen a weak measurement is made at B or C. Howeverwe see that it is the linear amplitude at E that is respon-sible for the experimentally observable quantities P b and P c in the linear order.Up to now, we proposed a detailed design of mak-ing a weak measurement at location C to show that theessence of the weak measurement is not different fromusual quantum mechanics methods. More importantly,we show that, no matter how weak the measurement is,a measurement always disturbs the original system. Thisis the price we always need to pay in order to get the in-formation in an interferometric system of the type shownin Fig. 1. Thus we obtain contribution not only fromthe original system (from path A in our case) but alsothe perturbation coming form measurements we made (inpath B and/or C).In order to emphasize our point, we can consider an-other measurement on the same system at the same time.For example, we can place the meter atoms at positions Cand E and carry out the joint measurement. If Vaidmanis correct [18, 19], the pointer at E should not find anyphoton. However, even when we do the calculation thatfollows the logic of weak measurement, the correspondingjoint weak value yields a different result in accordancewith the usual quantum mechanical expectation. Thejoint weak value [23] corresponding to a click at D isgiven by h ψ f | U L | i h | U L | i h | U L U L | ψ i ih ψ f | U L U L U L U L | ψ i i = t r (13)This value is non-zero and is equal to A C . The reason isthat if the photon is found at C (and finally causes D to click), it must pass through path E. IV. CONCLUDING REMARKS
In summary, we have considered a generic system ofmeasurement of weak values in a nested Mach-Zehnderinterferometer and have shown, via a straightforwardquantum mechanical calculation, that the weak measure-ment and the subsequent evolution are consistent withour quantum mechanical expectations. There is no mys-tery or paradox in the simple set up of Fig. 1. The quan-tum mechanical paradigm that a measurement disturbsthe system can explain the outcome of a potential experi-ment. We note that we considered a particular model forthe system-meter interaction to illustrate our results butsimilar conclusions can be drawn in other systems such asin [12]. We also mention that the three-box paradox [2]corresponds to the special case r = 1 / √ t = p / ACKNOWLEDGMENTS
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