Statistics of nondemolition weak measurement
A measurement consists in coupling a system to a probe and reading the output of the probe to gather information about the system. The weaker the coupling, the smaller the back-action on the system, but also the less information conveyed. If the system undergoes a second measurement, the statistics of the first output can be conditioned on the value of the second one. This procedure is known as postselection. A postselected weak measurement of an observable can give a large average output of the probe when the postselected state is nearly orthogonal to the initial state of the system. This large value is an interference effect in the readout of the probe, which is initially in a coherent superposition of readout states (also known as pointer states). Usually, the weak interaction between system and probe is considered instantaneous, so that the dynamics of the probe can be neglected. However, for a weak measurement in solid-state devices, an interaction of finite duration is likely needed.
Here we show how this finite duration generates a contribution of the dynamical phase to the readout statistics. Furthermore, we derive interpolation formulas that are able to describe the statistics of the weak measurement for the whole range of pre- and postselected states. Phase-space averages appear in the expansion, suggesting an interpretation in terms of non-positive probabilities. Decoherence in the probe is also accounted for and it is pointed out the existence of a regime of intermediate coupling strength in which coherent oscillations can be observed in the probability of the readout.
Sequential or joint nonprojective measurements can reveal the quantum nature of the detectors. For example, the pioneering work of Arthurs and Kelly Arthurs and Kelly Jr. (1965) demonstrated how two weak measurements of momentum and position could be carried out simultaneously, and how the uncertainty principle obeyed by the detectors (or probes) showed up in their mutual back-action. While the theory of positive operator-valued (POV) measures (of which the weak and the strong measurements are particular cases) has its mathematical roots in the seminal papers of Neumark Neumark (1940, 1943) and has been developed since the 70s Davies (1976); Helstrom (1976); Holevo (1982), only in 1988 was it realized that a weak measurement followed by a strong one could lead to arbitrarily large values of the average output Aharonov et al. (1988), provided that the weak measurement is conditioned on the result of the strong one. This conditioning is called postselection. As postselected weak measurements can give an arbitrarily large average output (even for non-orthogonal preparation and postselection Di Lorenzo (2012a)), they have been used to amplify a weak signal Hosten and Kwiat (2008); Dixon et al. (2009); Starling et al. (2009), to settle fundamental issues — such as determining the traversal time of a barrier Steinberg et al. (1993), solving Hardy’s paradox Yokota et al. (2009); Lundeen and Steinberg (2009), and observing quantum trajectories Kocsis et al. (2011)— and also to perform quantum state tomography Lundeen et al. (2011). Concerning the last application, there are several proposals Di Lorenzo (2011); Haapasalo et al. (2011); Lundeen and Bamber (2012); Di Lorenzo (2012b) for extending the procedure to mixed states. It has also been shown Di Lorenzo and Nazarov (2004); Di Lorenzo et al. (2006) that having a stream of particles sent to probes initially prepared to measure in the strong regime can create coherence in the probes; this drives the measurement to the weak regime and is reflected in a deviation of the statistics from that predicted for projective measurement, provided the decoherence rate of the probes does not exceed the firing rate of the particles. For other applications and studies of the weak measurement, see Ref. Kofman et al. (2012).
The theoretical papers on postselected weak measurements are mostly limited to the study of the average value and assume an instantaneous (von Neumann) interaction. While for optical implementations postulating an instantaneous interaction is reasonable, for the still prospective realizations in solid-state systems (see the proposals Williams and Jordan (2008); Romito et al. (2008); Shpitalnik et al. (2008); Ashhab et al. (2009a, b); Bednorz and Belzig (2010); Zilberberg et al. (2011)) it is more realistic to suppose a coupling that lasts a finite time. In an earlier paper Di Lorenzo and Egues (2008), we introduced a finite-duration interaction, assuming only that the measurement was a quantum nondemolition one Braginsky and Khalili (1992). The results of Ref. Di Lorenzo and Egues (2008), however, were limited to the average value of the output and its variance (barring the case study of a spin 1/2, for which, since an exact solution exists, the expansion in the weak coupling was made merely to test its validity). In a more recent paper Di Lorenzo (2012c), one of us studied the whole statistics of the weak measurement, by performing a controlled expansion in the coupling strength, as in Ref. Wu and Li (2011), and providing an interpolation formula that works for any preparation and postselection. Here, we provide the statistics of a nondemolition weak measurement of an arbitrary variable. We show that the finite duration has observable consequences, since in a weak measurement the coherence of the probe manifests through the contribution of its density matrix off-diagonal elements (in the readout basis), and since the interaction contributes dynamical phases to these elements.
Ii Description of the measurement.
Let us consider a quantum system prepared, at time , in a state (preselection) and a second quantum system, the probe, prepared, at time , in a state . Let the Hamiltonian of the system be , and that of the probe . The system and the probe interact, at time , through , where is an operator on the system’s Hilbert space, on the probe’s, and is a function that vanishes outside a finite interval , with . The interaction generates a time-evolution operator that acts on the Hilbert space of the system and the probe, entangling the two. Let the operator be the conjugate observable 111While we are using the notation for the write-in and readout variables, these need not be literal position and wave number. Many authors use , , . of , . It follows from Heisenberg’s equations of motion, or even from Hamilton’s equations for the corresponding classical case, that is the observable of the probe that carries information about the measured quantity . As the interaction lasts a finite time, has to be be conserved, i.e. , otherwise the question would arise of what is being measured 222If, instead, we would assume an instantaneous interaction, , , then we could allow .. We notice that, in order for the measurement to be nondemolition, too must be conserved during the free evolution of the probe and change only due to the interaction with the observed system. Hence . In other words, we are considering not a von Neumann (instantaneous) weak measurement, but the more general nondemolition (finite-duration) weak measurement, having the former as a special limiting case. At time a projective measurement of the observable is made on the probe. As is conserved by , the value obtained will not depend on the time, as long as . In a strong nondemolition measurement, it is furthermore assumed that the probe is prepared in a state such that the elements are negligible unless , where the function maps the values of to the readout. This hypothesis guarantees the fidelity of the measurement, as well as its repeatability. However, as we are interested in the weak regime, we drop this assumption.
Iii Postselection in a mixed state.
At time , a projective measurement of an observable of the system is made, giving an output and leaving the system in the state . Then, given , one keeps the outcome according to an arbitrarily chosen probability . This step leaves the system in the postselected mixed state
The procedure detailed above and sketched in Fig. 1, describes a measurement with pre- and postselection. Without loss of generality, we shall consider and , with the duration of the interaction.
We notice that is not necessarily 1 as are probabilities conditional on the event , not probabilities of the event. If the system has a finite-dimension Hilbert space, with the dimension, then . For if and 0 otherwise, the postselection is in the pure state . In this case, is a sub-optimal choice, in the sense that some of the trials are discarded unnecessarily. The opposite limit is found for . This implies that no postselection is made, i.e., . The optimal choice is . If all the probabilities are multiplied by the same factor , , the postselected state is unchanged, but the probability of a successful postselection is also multiplied by , and so is the joint probability of observing the probe in and successfully post-selecting the system. Thus the conditional probability, given by the ratio of the two probabilities above, is unaffected by this rescaling. The optimal choice of is such that it maximizes the probability of postselection, namely . Hence , with the equality only for postselection in a pure state. In a sense, we could say that the probabilistic postselection leaves the system in the unnormalized state .
Another proposed method Wiseman (2002) is to make the postselecting measurement a POV one. This way is replaced by a positive operator . However, while appears in the probabilities, as e.g. in Eq. (5) below, the system is not selected in a state but in , with its reduced density matrix after the interaction with the probe.
Iv Exact results.
The joint state for the probe and the system, at any time is
where is the time evolution operator generated by , are the simultaneous eigenstates of and , and are the eigenvalues of corresponding to . Following a lemma demonstrated in the Supplemental Material section, there is an analytic solution for the propagator,
where we define the Hamiltonian phase
For a von Neumann measurement, .
The conditional probability of observing the outcome , given that the state has been postselected in , follows from Bayes’s rule,
where the denominator represents the probability of making a successful postselection in , irrespective of the value of , or of what observable of the probe, if any, was measured. What remains to be done is simple: apply the controlled expansion of both and in as in Ref. Di Lorenzo (2012c), with the difference that here one should keep track of additional contributions from the dynamical phases . It is fundamental to keep in mind that, both in and in , the zeroth and first order terms vanish for nearly orthogonal pre- and postselected states (NOPPS). Thus one should write down as the ratio of two quadratic polynomials in , without succumbing to the temptation to expand the denominator.
Before proceeding to the expansion, we introduce the normal weak values as in Ref. Di Lorenzo (2012c)
We note that , so that are real. For NOPPS, and , with for ; also, for . While there can be exceptions to this behavior when for some integer , in any case stays finite for NOPPS, barring some trivial instances. This consideration is important, as provides the dominant term to both and for NOPPS.
We shall also use the phase-space averages Moyal (1949), , with the initial Wigner function of the probe. In particular, we define the covariance . A natural concept arising in the expansion is that of phase-space conditional averages , defined by
with . The phase-space average of is found by integrating Eq. (8) over . Finally, we define the time scales , which satisfy .
Vi Weak compared to what?
As is a dimensionful constant, it has to be compared to some homogeneous quantity in order to establish whether the measurement is weak, strong, or intermediate. Let be the coherence scale, i.e., , and the classical uncertainty scale, i.e., . Precisely, we define , where is the classical uncertainty of the conjugate variable of , , and . The uncertainty relation requires . In order to realize the weak (coherent) regime, the coupling constant must be small compared to the coherence scale, precisely, , with the maximum distance between the eigenvalues of . The validity of the expansion relies also Wu and Li (2011) on . Equation (5) shows that both the off-diagonal terms of the detector and the Hamiltonian phase contribute to the statistics. Thus we are in presence of interference. We also remark that if either or , i.e., no postselection or no preselection is made, the coherent contributions disappear. Hence, both pre- and postselection are essential in order for interference to show up.
Vii Controlled expansion.
The probability of postselection is found by expanding the integral of Eq. (5) (for simplicity, ),
where and the prime stands for differentiation. Notice that is the velocity, so that is the displacement of the variable. Due to its non-trivial dynamics as , and hence the velocity, changes during the interaction with the system, appears instead of 333For simplicity, consider a constant interaction, so that varies linearly in time for fixed (uniformly accelerated motion). Then , and the classical displacement is indeed . Notice that the appearing in the formulas refers to the value at time , hence is the final velocity .. Since the second-order term contributes significantly only for NOPPS, we have neglected compared to .
Furthermore, as all expressions are homogeneous in , we may reduce the number of independent parameters by dividing by, say, . Accordingly, we introduce the canonical complex weak value and the additional real term . We remark that . The equality holds whenever and are mixtures of pure states and each pair of which has the same weak value . In particular, for pure pre- and postselected states.
It is convenient to introduce the characteristic function , from which the moments of the distribution can be generated as , with average over . By integrating Eq. (5) times , changing the variable of integration to , expanding, and normalizing,
The interpolating formula for the probability is instead
The derivatives come from having expanded . This way, the shifts of the probability distribution are lost. However, as in the weak regime the shifted peaks are not resolved, this yields a small deviation from the exact result.
As an example, away from NOPPS, we can expand the average value of to first order,
The contribution from is usually neglected, as in an instantaneous measurement and as a rule and are assumed to be initially uncorrelated. The correlator is proportional to the derivative of the variance of under specific hypotheses Jozsa (2007).
Furthermore, it may be useful to observe the value of the probe, instead of . We give the full statistics in the Supplemental Materials section, and give here the average value of , which is, in the regime away from NOPPS, where we are allowed to keep up to first order terms,
The results of Ref. Jozsa (2007) are recovered as a special case.
As an effective model for decoherence, we consider a random classical force acting on the probe, while the latter interacts with the system. In principle, if the probe is prepared before the interaction starts, decoherence will tend to kill the off-diagonal elements , driving the measurement to the weak incoherent regime. However, this effect can be counteracted by preparing the probe shortly before its interaction with the system, and in any case it can be treated rather simply. On the other hand, decoherence during the interaction with the system is unavoidable, as the probe is open to external influences, which can come from the system and from the environment. For simplicity, we take . The effect consists in the addition of an imaginary phase to in Eq. (5), which becomes
We assumed the random force to have the correlator , with a constant, the temperature, and Boltzmann’s constant, and we defined the decoherence rate 444Strictly speaking, as is not necessarily a wave number, does not have the dimensions a mass, and hence does not have the dimensions of a rate.. The factor is a dimensionless parameter that depends on the details of . Thus there is a new scale for to be compared with, the decoherence scale . If , the contribution from the off-diagonal elements of becomes negligible, and we fall into the weak incoherent regime. If , the net result of decoherence is simply to shift the coefficients of in Eqs. (10) and (12).
Ix A case study.
We consider a spin-1/2 system preselected in a state , on which a weak measurement of is made, and that is postselected in . The interaction is considered constant and the probe is prepared in . The free Hamiltonian of the probe is as in the previous section . This defines the Hamiltonian scale . We show the approximate and exact probability in Fig. 2.
In the intermediate case , it is no longer legitimate to expand . Then, as shown in Fig. 3, the probability displays coherent oscillations in .
We have studied the statistics of a weak nondemolition measurement, providing expressions for the probability and the characteristic functions that are robust for any overlap between the postselection and the preparation, contrary to the results of Ref. Aharonov et al. (1988) and subsequent papers on weak measurement. We have included decoherence in an effective, albeit heuristic, way, and we have pointed out the existence of a regime of intermediate strength, in which coherent oscillations can be observed.
Acknowledgements.A.D.L. performed this work as part of the Brazilian Instituto Nacional de Ciência e Tecnologia para a Informação Quântica (INCT–IQ) and was supported by Fundação de Amparo à Pesquisa do Estado de Minas Gerais through Process No. APQ-02804-10.
- Arthurs and Kelly Jr. (1965) E. Arthurs and J. L. Kelly Jr., Bell System Tech. J. 44, 725 (1965).
- Neumark (1940) M. A. Neumark, Izv. Akad. Nauk SSSR, Ser. Mat. 4 53 (1940).
- Neumark (1943) M. A. Neumark, C. R. (Doklady) Acad. Sci. URSS (N.S.) 41 (1943).
- Davies (1976) E. B. Davies, Quantum Theory of Open Systems (Academic Press, London, 1976).
- Helstrom (1976) C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, London, 1976).
- Holevo (1982) A. S. Holevo, Probabilistic and statistical aspects of quantum theory (North Holland, Amsterdam, 1982).
- Aharonov et al. (1988) Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).
- Di Lorenzo (2012a) A. Di Lorenzo, preprint arXiv:1210.3274 (2012a), to appear in Phys. Rev. A.
- Hosten and Kwiat (2008) O. Hosten and P. Kwiat, Science 319, 787 (2008).
- Dixon et al. (2009) P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Phys. Rev. Lett. 102, 173601 (2009).
- Starling et al. (2009) D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, Phys. Rev. A 80, 041803 (2009).
- Steinberg et al. (1993) A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, Phys. Rev. Lett. 71, 708 (1993).
- Yokota et al. (2009) K. Yokota, T. Yamamoto, M. Koashi, and N. Imoto, New Journal of Physics 11, 033011 (2009).
- Lundeen and Steinberg (2009) J. S. Lundeen and A. M. Steinberg, Phys. Rev. Lett. 102, 020404 (2009).
- Kocsis et al. (2011) S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
- Lundeen et al. (2011) J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, Nature 474, 188 (2011).
- Di Lorenzo (2011) A. Di Lorenzo, Phys. Rev. A 83, 042104 (2011).
- Haapasalo et al. (2011) E. Haapasalo, P. Lahti, and J. Schultz, Phys. Rev. A 84, 052107 (2011).
- Lundeen and Bamber (2012) J. S. Lundeen and C. Bamber, Phys. Rev. Lett. 108, 070402 (2012).
- Di Lorenzo (2012b) A. Di Lorenzo, preprint arXiv:1205.1238 (2012b).
- Di Lorenzo and Nazarov (2004) A. Di Lorenzo and Y. V. Nazarov, Phys. Rev. Lett. 93, 046601 (2004).
- Di Lorenzo et al. (2006) A. Di Lorenzo, G. Campagnano, and Y. V. Nazarov, Phys. Rev. B 73, 125311 (2006).
- Kofman et al. (2012) A. G. Kofman, S. Ashhab, and F. Nori, Physics Reports 520, 43 (2012).
- Williams and Jordan (2008) N. S. Williams and A. N. Jordan, Phys. Rev. Lett. 100, 026804 (2008).
- Romito et al. (2008) A. Romito, Y. Gefen, and Y. M. Blanter, Phys. Rev. Lett. 100, 056801 (2008).
- Shpitalnik et al. (2008) V. Shpitalnik, Y. Gefen, and A. Romito, Phys. Rev. Lett. 101, 226802 (2008).
- Ashhab et al. (2009a) S. Ashhab, J. Q. You, and F. Nori, Phys. Rev. A 79, 032317 (2009a).
- Ashhab et al. (2009b) S. Ashhab, J. Q. You, and F. Nori, New Journal of Physics 11, 083017 (2009b).
- Bednorz and Belzig (2010) A. Bednorz and W. Belzig, Phys. Rev. Lett. 105, 106803 (2010).
- Zilberberg et al. (2011) O. Zilberberg, A. Romito, and Y. Gefen, Phys. Rev. Lett. 106, 080405 (2011).
- Di Lorenzo and Egues (2008) A. Di Lorenzo and J. C. Egues, Phys. Rev. A 77, 042108 (2008).
- Braginsky and Khalili (1992) V. B. Braginsky and F. Y. Khalili, Quantum Measurement (Cambridge University Press, Cambridge, 1992).
- Di Lorenzo (2012c) A. Di Lorenzo, Phys. Rev. A 85, 032106 (2012c).
- Wu and Li (2011) S. Wu and Y. Li, Phys. Rev. A 83, 052106 (2011).
- Wiseman (2002) H. M. Wiseman, Phys. Rev. A 65, 032111 (2002).
- Moyal (1949) J. E. Moyal, Mathematical Proceedings of the Cambridge Philosophical Society 45, 99 (1949).
- Jozsa (2007) R. Jozsa, Phys. Rev. A 76, 044103 (2007).
Appendix A A useful lemma
We prove the following lemma: for a system subject to the time-dependent Hamiltonian , with arbitrary functions and , with the time-evolution being hence , the propagator reads
a.1 Brute force derivation
We use the path-integral technique, but without path-integrals, i.e. we approximate the time-ordered exponential as a product of terms, and we introduce between each term the identity in the basis, obtaining ()
In the limit , the sums in the last line of Eq. (17) become integrals, and the approximated equality with the first line becomes exact, thus the lemma is proved.
a.2 Alternative derivation
We provide an alternative, more elegant derivation of the above result. The technique illustrated below may find applications in other fields. The Schrödinger equation in wave number space is
Upon rearranging the terms and dividing by , we have the non-homogeneous first order partial differential equation
with . The solution to the corresponding homogeneous equation is , with an arbitrary function of one variable.555We may change the lower limit of integration to an arbitrary constant , but this gives an additional constant in the argument of that can be reabsorbed in the arbitrary function . Thus we chose zero as lower limit, just because we like it. This suggests to change the variables to , with . Let . The PDE becomes then
and a particular solution is readily found
so that, in terms of the original function, the general solution is
The arbitrariness of can be exploited to find the solution of Eq. (18) with the initial condition :
Appendix B Statistics of the write-in variable
The characteristic function is
with , , , . The normalization is given by . We note that is the characteristic function for that the probe would have after a time , had it not interacted with the system.
Appendix C Measurement of a spin 1/2
c.1 Exact expressions
We use units of , , , . We assume the free Hamiltonian for the probe to be . This position defines the dynamical scale: , with .
For definiteness , with the coherence scale. We let , , , with a unit vector and , . The probability of a successful postselection is then
The joint probability is
The characteristic function is
c.2 Coherent oscillations
Equation (27) shows that the off-diagonal terms of provide oscillating terms, with a period . If exceeds the scale over which decays, the oscillations can not be discerned. If instead , coherent oscillations are observed. In this case, we can not expand in the exponential of the Hamiltonian phases , but should retain the full terms. This can be implemented through the transformation , with the unitary operator
For a spin 1/2, is but a rotation around the direction of . Accordingly, the weak value becomes a periodic function of
Furthermore and in Eqs. (10)-(14). Once these prescriptions are applied, the probability in Eq. (13) approximates excellently the exact expression.