Reflection of a Particle from a Quantum Measurement
RReflection of a Particle from a QuantumMeasurement
Jonathan B. Mackrory , Kurt Jacobs , Daniel A. Steck Department of Physics and Oregon Center for Optics, 1274 University of Oregon,Eugene, OR 97403-1274 Department of Physics, University of Massachusetts at Boston, 100 Morrissey Blvd,Boston, MA 02125, USAPACS numbers: 03.65.Ta, 37.10.Vz, 02.50.Ey, 42.50.Lc
Abstract.
We present a generalization of continuous position measurements thataccounts for a spatially inhomogeneous measurement strength. This describes manyreal measurement scenarios, in which the rate at which information is extracted aboutposition has itself a spatial profile, and includes measurements that detect if a particlehas crossed from one region into another. We show that such measurements canbe described, in their averaged behavior, as stochastically fluctuating potentials ofvanishing time average. Reasonable constraints restrict the form of the measurementto have degenerate outcomes, which tend to drive the system to spatial superpositionstates. We present the results of quantum-trajectory simulations for measurementswith a step-function profile (a “which-way” measurement) and a Gaussian profile. Wefind that the particle can coherently reflect from the measurement region in both cases,despite the stochastic nature of the measurement back-action. In addition, we explorethe connection to the quantum Zeno effect, where we find that the reflection probabilitytends to unity as the measurement strength increases. Finally, we discuss two physicalrealizations of a spatially varying position measurement using atoms. a r X i v : . [ qu a n t - ph ] S e p eflection of a Particle from a Quantum Measurement Contents1 Introduction 22 Equations of Motion 4
Position measurements strike at the heart of what distinguishes quantum mechanicsfrom classical mechanics. Within classical mechanics, it is possible to localize particleswith arbitrarily fine precision for all times. In quantum mechanics, the back-action fromthe measurement disturbs the particle and implies that there are fundamental limits tomeasurement precision—limits that are being approached in current experiments [1, 2].The usual formalism of projective measurements fails when applied to positionmeasurements: if a particle is reduced to an eigenstate of position it has an infinitemomentum uncertainty and hence infinite energy. Position measurements are bestdescribed by considering the “weak” limit where the observer continuously extractsinformation about the particle’s location at a given rate; in a small time interval only asmall amount of information is extracted.Continuous measurements for quantum systems are usually phrased in the languageof stochastic master equations, where the system’s density matrix is updated based eflection of a Particle from a Quantum Measurement position measurements in particular can be employed in feedback control loops for coolingquantum systems [11]. This has been applied to atoms in cavities [12–15], trappedions [16, 17], and nanomechanical resonators [18]. In addition to seeking to controlquantum systems, continuous position measurements also provide one path through thequantum-to-classical transition. In particular, continuously monitored quantum systemsare able to exhibit chaotic behavior [19, 20], in contrast to closed quantum systems.Considering the importance of position measurements, it is necessary to developthe theory to account for realistic constraints applicable to any experimental realizationof a position measurement. In particular, in any real position measurement the particlecan only be detected within a limited region. If the particle leaves this region, theobserver gains no further knowledge of the particle’s position. This can be modeledby a space-dependent coupling of the particle to a bath, such as the radiation field.When the strength of the coupling to the bath, and thus the measurement strength , isitself a function of the position, it is, in fact, this function of the position that becomesthe measured observable. The coupling correlates the bath with that function of theparticle’s position, and so monitoring the bath allows the observer to gather informationabout that function.The average dynamics of the measured particle (that is, the motion of the particleaveraged over all the possible measurement results) can be reproduced by a fluctuatingpotential with the same position dependence as the measurement. Thus, as far asthe average motion is concerned, the measurement acts like a stochastic force. Thisfluctuating potential provides an alternate, intuitive physical picture for understandingthe measurement back-action. We expand on this unitary “unraveling” of the averagemotion in Sec. 2.2.It has been shown previously by a number of authors that a measurement thatdetermines whether a particle is on one side of a dividing line or the other can excludethe wave-function from the region in which the particle is initially absent [21–29]. Thiscauses the particle to reflect from the dividing line, and is due to the quantum Zeno eflection of a Particle from a Quantum Measurement coherently reflect from the resonant light field for a sufficiently large intensity,despite the absence of a mean dipole force, and despite the stochastic nature ofresonant atom–light interactions that tend to heat the atom. This phenomenon, besidesbeing somewhat counterintuitive, has implications in situations where atoms encounterlocalized, resonant, optical-pumping fields—as occurs, for example, in implementationsof one-way barriers for atoms [30–33]. We present further details on this realization inSec. 5.1, along with another example of an atom interacting with an off-resonant cavity.We begin in the next section by deriving the stochastic master equation thatdescribes a continuous position measurement with a spatially varying measurementstrength, and elucidate some of its key properties. In Sec. 3 we perform simulationsof a particle incident on measurements with two kinds of spatial profiles, showing thebehavior of the particle on individual realizations of the measurement, which may involveeither reflection or transmission, as well as the ensemble-averaged behavior. In Sec. 4we discuss the quantum Zeno effect, in Sec. 5.1 we present two physical realizations ofa spatially varying measurement, and in Secs. 6 and 7 we finish with some concludingremarks.
2. Equations of Motion
In this section we will derive the equation of motion describing a spatially varyingposition measurement. Our measurements are usually made by monitoring a large baththat interacts irreversibly with the system. The position measurement arises from aspatially-dependent potential that couples the particle to the bath. The bath thenbecomes correlated with a real function of the particle’s position µ ( x ) and we candistinguish different positions by how strongly they interact with the bath. Although eflection of a Particle from a Quantum Measurement µ is real, we will keep our notation general because later we will add a complex local-oscillator amplitude to µ . We will start our derivation from the positive-operator-valuedmeasure for a measurement with two outcomes in each infinitesimal time-step, dt (notethat our treatment and superoperator notation parallels that of Ref. [34]). We will referto the two outcomes respectively as “no detection,” and “detection”. (We choose thisnomenclature due to the formal similarity to photodetection, though these outcomesare not necessarily tied to a photodetector.) The measurement operators that describethe two outcomes areΩ = 1 − i (cid:126) H dt − κµ † ( x ) µ ( x ) dt (1)Ω = √ κ dt µ ( x ) , (2)where H = p / (2 m ) + V ( x ) is the free Hamiltonian for the system, κ is the measurementstrength, Ω corresponds to “no detection”, and Ω to a “detection”. The detectionoutcome occurs with probability Tr[Ω † Ω ρ ] = 2 κ (cid:104) µ † ( x ) µ ( x ) (cid:105) dt. The evolution thatoccurs between detections is due to Ω ,ρ → Ω ρ Ω † Tr[Ω † Ω ρ ] = ρ − i (cid:126) [ H, ρ ] dt − κ H [ µ † ( x ) µ ( x )] ρ dt, (3)where the H super-operator is defined as H [ c ] ρ = cρ + ρc † − (cid:104) c + c † (cid:105) ρ. (4)If detection occurs the density matrix changes according to ρ → Ω ρ Ω † Tr[Ω † Ω ρ ] = µ ( x ) ρµ † ( x )Tr[ µ † ( x ) µ ( x ) ρ ] . (5)We can represent all of this evolution in a single stochastic master equation (SME): dρ = − i (cid:126) [ H, ρ ] dt − κ H [ µ † ( x ) µ ( x )] ρ dt + G [ µ ( x )] ρ dN, (6)where G [ c ] ρ = cρc † Tr[ c † cρ ] − ρ. (7)Here, dN is a Poisson process [35, 36], where dN = 1 with probability 2 κ (cid:104) µ † ( x ) µ ( x ) (cid:105) dt and is zero otherwise. If we mix in a “local oscillator” with this signal, then we passover to a master equation similar in form to the usual position-measurement masterequation [8]. This corresponds to the transformation µ → µ + α √ κ (8 a ) H → H − i (cid:126) √ κ α ∗ µ − αµ ) , (8 b )where α = | α | e iφ is the complex amplitude of the local oscillator, which leavesthe unconditioned evolution (that is, the evolution averaged over all the possiblemeasurement results) unchanged. For large | α | , we can pass over to the white noise limit. eflection of a Particle from a Quantum Measurement dN by a mean drift and Gaussian fluctuations, dN = (cid:28)(cid:28) dNdt (cid:29)(cid:29) dt + (cid:115)(cid:28)(cid:28) dNdt (cid:29)(cid:29) dW, (9)where double angle brackets denote the ensemble average for the random variable,and dW is Ito white noise, with (cid:104)(cid:104) dW (cid:105)(cid:105) = 0 and dW = dt [35, 36]. We apply thetransformations in Eqs. (8 a ), (8 b ), and the white noise limit of Eq. (9), to the jumpmaster equation (6). In the large | α | limit, the resulting white noise master equation is dρ = − i (cid:126) [ H, ρ ] dt + 2 κ D [ µ ( x )] ρ dt + √ κ H [ e − iφ µ ( x )] ρ dW, (10)where the D superoperator is defined as D [ c ] ρ = cρc † − (cid:0) c † cρ + ρc † c (cid:1) . (11)For φ = 0, Eq. (10) is simply the standard stochastic master equation that describes themeasurement of the observable O = µ ( x ) [8]. Thus the case µ ( x ) = x recovers the usualmaster equation for a measurement of position. The continuous stream of measurementresults, often referred to as the measurement record , is given by dr = (cid:104) µ ( x ) (cid:105) dt + dW √ κ , (12)where dr is the result in the time interval dt .The unconditioned dynamics are given by the master equation ∂ t ρ = − i (cid:126) [ H, ρ ] − κ [ µ ( x ) , [ µ ( x ) , ρ ]] , (13)where we have taken an ensemble average over all possible noise realizations. An alternative interpretation is suggested if we choose a different phase for the localoscillator. If we instead choose φ = π/
2, the master equation becomes dρ = − i (cid:126) [ H dt + √ κ (cid:126) µ ( x ) dW, ρ ] + 2 κ D [ µ ( x )] ρ dt, (14)= − i (cid:126) [ H dt + √ κ (cid:126) µ ( x ) ◦ dW, ρ ] , (15)where the first equation is in Ito form, and in the second equation ◦ dW denotesStratonovich white noise [35, 36]. This phase choice corresponds to measuring the otherbath quadrature [37], yielding information about the measurement backaction ratherthan position. The effect of the bath here is formally equivalent to a stochastic potential.The spatial dependence µ ( x ) of the coupling to the bath determines the profile of thestochastic potential. The term in D [ µ ( x )] is the Ito correction to the Stratonovichfluctuating potential. When averaged over all noise realizations, this master equationrecovers the unconditioned evolution given by Eq. (13). eflection of a Particle from a Quantum Measurement H SB dt = µ ( x )( dB in + dB † in ) , (16)where dB in is a white noise bath operator due to the sum over bath modes, all evolvingat different frequencies [39]. This interaction term correlates the state of the bath with µ ( x ). The key observation is that since the interaction Hamiltonian is proportionalto µ ( x ), the noise in the bath drives the system through µ ( x ). The bath thereforeexerts a stochastic force on the particle whenever µ ( x ) is inhomogeneous. Note that any measurement of µ ( x ) that is mediated by a bath must have an interaction of thisform [40]. This in turn implies that there must be a stochastic force of the same form.As a concrete example, a two-level atom interacting with a resonant laser field feels thestochastic dipole force due to spontaneous emission [41]. We can gain insights into the dynamics induced by a spatially varying positionmeasurement by examining the lowest few terms in the Taylor expansion of µ ( x ) aboutthe current mean position (cid:104) x (cid:105) . If µ ( x ) is constant then the post-measurement state iscompletely unaffected, and the measurement does nothing.The linear term in the Taylor expansion acts as a standard position measurement.The measurement can then be approximated by √ κµ ( x ) ≈ √ κµ (cid:48) ( (cid:104) x (cid:105) )( x − (cid:104) x (cid:105) ) , (17)which defines the effective local measurement strength, where µ (cid:48) ( x ) ≡ dµ/dx . Theconstant (cid:104) x (cid:105) term again has no effect. This linear approximation is valid if the particleis well-localized on the scale of the measurement function. This relation provides thelink between a general, inhomogeneous measurement of position and the usual, linearposition measurement.The quadratic term in the Taylor expansion is important at the maxima and minimaof the measurement function where the linear term vanishes. The effective measurementis √ κµ ( x ) ≈ √ κµ (cid:48)(cid:48) ( (cid:104) x (cid:105) )( x − (cid:104) x (cid:105) ) , (18)which drives the state to spatial superpositions since the measurement result cannotdistinguish (cid:104) x (cid:105) + x from (cid:104) x (cid:105) − x [42]. Once the components of the superposition moveaway from the extremum, the local measurement for each component becomes linear.Each component will then evolve under their respective effective measurements. We now derive the conditions that the measurement function µ ( x ) must satisfy for it tocorrespond to a position measurement that acts only in a limited region of space. First eflection of a Particle from a Quantum Measurement (a) m o o ( x ) x (b) m o o ( x ) x (c) m o o ( x ) x Figure 1.
Possible measurement functions realizing a position measurement within abounded region. The local measurement strength is proportional to the derivative of µ ( x ). (a) Measurement function with different asymptotic values. (b) Measurementfunction with a discontinuous return to zero. (c) Continuous measurement functionwith same asymptotic values. we expand the measurement terms in the master equation (10) in the position basis,with ρ ( x, x (cid:48) ) = (cid:104) x | ρ | x (cid:48) (cid:105) : dρ ( x, x (cid:48) ) = − κ [ µ ( x ) − µ ( x (cid:48) )] ρ ( x, x (cid:48) ) dt + √ κ [ µ ( x ) + µ ( x (cid:48) ) − (cid:104) µ (cid:105) ] ρ ( x, x (cid:48) ) dW. (19)The leading term leads to decay of coherence between positions x and x (cid:48) where µ differ.If we enforce µ ( x ) = µ ( x (cid:48) ), then the measurement cannot distinguish between x and x (cid:48) .This implies that µ ( x ) should be constant outside the measurement region.Inside the measurement region, the measurement function µ ( x ) should be linearto act as a standard position measurement. However, if µ ( x ) does not return to thesame value on either side of its linear section (as in Fig. 1(a)), then the measurementwould collapse superpositions that are on either side of, and completely outside thelinear measurement region. This is not what one normally means when restricting ameasurement to a particular region. For example, a photodetector of some finite sizeplaced well within the arms of an optical interferometer should not collapse the fringes bygiving which-way information, but this is precisely the case for the function in Fig. 1(a).Thus, for a measurement to truly act only within some bounded region, the functionmust take on the same value everywhere outside that region. We may take this valueto be zero, since shifting µ ( x ) by a constant has no effect on the dynamics.Generally speaking, µ ( x ) will be the profile of some continuous (e.g., laser) field.Thus, a discontinuous return of the measurement function to the external value, as inFig. 1(b), is at best an idealization of a more physical, continuous measurement function.Along the same lines, the measurement should also be finite in extent. Measurementfunctions of the type in Fig. 1(a) must therefore also be idealizations of measurementfunctions that are eventually zero, though possibly only returning to zero far away fromthe location of the measured particle.The strictly physical form of the measurement function must therefore becontinuous and return to the same constant value on both sides of the measurementregion, as shown in Fig. 1(c). In particular, this means that there will always beindistinguishable positions x , x , where µ ( x ) = µ ( x ) within the measurementregion, corresponding to the same measurement outcome. At any given instant, the eflection of a Particle from a Quantum Measurement x , x , even if the measurementis not symmetric on either side of the extremum, for example if µ (cid:48) ( x ) (cid:54) = µ (cid:48) ( x ). Themeasurement will thus create a superposition. However, if we also account for theHamiltonian evolution then the backaction from an asymmetric measurement can givethe components of the superposition differing momenta, and hence different positions atlater times. Hence over an extended period of time the measurement could distinguishbetween the two positions. Likewise, we have argued that for a single measurementchannel, a strictly physical measurement function must result in positions that areindistinguishable. However, multiple position-measurement channels—each with theirown ambiguities—may be combined to remove all the ambiguities. These are examplesof combining multiple measurements to gain information that cannot be gleaned fromeither measurement separately. A more familiar example is that information fromposition measurements at two different times may be combined to obtain informationabout momentum. We can derive equations of motion for the two lowest-order moments of the momentumprobability density based on Eq. (10). If we assume the particle Hamiltonian is givenby H = p m + V ( x ) , (20)we obtain d (cid:104) p (cid:105) = − (cid:104) ∂ x V (cid:105) dt + √ κ [ (cid:104) µp + pµ (cid:105) − (cid:104) µ (cid:105)(cid:104) p (cid:105) ] dW (21) d (cid:104) p (cid:105) = − (cid:104) p∂ x V + ∂ x V p (cid:105) dt + 2 (cid:126) κ (cid:104) ( ∂ x µ ) (cid:105) dt + √ κ [ (cid:104) µp + p µ (cid:105) − (cid:104) µ (cid:105)(cid:104) p (cid:105) ] dW. (22)In an ensemble average d (cid:104)(cid:104) p (cid:105)(cid:105) = −(cid:104) ∂ x V (cid:105) dt , so the average momentum is not changed bythe measurement. For a free-particle ensemble, the average momentum is conserved—during a single trajectory, the measurement can exert a force, but the force vanisheson average. The measurement back-action enters as diffusion in the ensemble-averagedmoment evolution, d (cid:10)(cid:10) p (cid:11)(cid:11) = 2 (cid:126) κ (cid:104) ( ∂ x µ ) (cid:105) dt, (23)with momentum diffusion coefficient D p = 2 (cid:126) κ (cid:104) ( ∂ x µ ) (cid:105) . This reflects that it is the gradient in µ ( x ) that gives relative position information.
3. Simulations
We now present some quantum-trajectory simulations that reveal some of the noveldynamics that occur under inhomogeneous position measurements. In the simulations,the particle starts in a minimum-uncertainty, Gaussian state with mean position (cid:104) x (cid:105) = x , mean momentum (cid:104) p (cid:105) = p and root-mean-square width σ x . It is incident eflection of a Particle from a Quantum Measurement µ ( x ) from outside the measurement region, from the left-hand side. Since the atom starts in a pure state we initially have maximal knowledgeabout the atom. Our results focus on the case when p /σ p (cid:29)
1, which corresponds toa well-defined momentum. In the limit of small κ , the particle simply passes throughthe measurement region with mild heating, so we also restrict ourselves to the regimeof strong measurements, κ > p / (2 m (cid:126) ).Our simulations use a split-operator Fourier method to propagate the stochasticSchr¨odinger equation associated with Eq. (10). We have set (cid:126) = m = 1 throughout ourpresentation here. We visualize the evolution of the particle under the measurementusing the Wigner function for the particle [43]. We will first consider a step measurement function µ ( x ) = θ ( x ) = (cid:110) x > x < , (24)which is simple and analytically tractable. It also affords a comparison with a familiarexample from textbook quantum mechanics: the potential step. This corresponds tothe limit of an arbitrarily large gradient for a continuous measurement. This limit isunphysical, strictly speaking, but it is useful as an idealization of a strong measurementover a small region.Fig. 2 shows the temporal evolution of a wave packet incident on a static potentialstep V ( x ) = V θ ( x ), with no measurement, as a context for understanding themeasurement-driven evolution. The evolution can be described by the Schr¨odingerequation, or by setting κ = 0 in the measurement master equation (10), and includingthe potential V ( x ). The height of the potential barrier in this case is smaller than thekinetic energy of the wave packet, so the particle can transmit over the barrier. The wavepacket splits into two pieces, reflected and transmitted. The reflected wave packet hasexactly the opposite momentum of the inbound wave packet. However, the transmittedwave packet is slower, since the particle lost energy climbing the potential step. Thestriated red and blue regions denote interference. The fringes between the reflectedand transmitted wave packets show that they represent a coherent superposition. Notethat the fringes between the two coherent components are oriented along the directionbetween them. This corresponds to the fact the fringes would yield an interferencepattern in the appropriate marginal probability distributions [43]. Missing fringesindicate a classical mixture of the two states.Fig. 3 shows the evolution of a wave packet incident on a measurement step µ ( x ) = θ ( x ), but for a free particle with no explicit potential. The step measurementdistinguishes whether the particle is on the left- or the right-hand side of the origin, butit does not resolve different positions on a given side. This means that superpositionsconfined to one side are unaffected by the measurement, while superpositions on differentsides of the origin will collapse. This measurement is analogous to a “which-way” eflection of a Particle from a Quantum Measurement p -3-2-10123 x -20 -10 0 10 20 t=2 t=4 t=6 t=8 V ( x ) / V Figure 2.
Wigner functions for a particle incident on a static potential step withheight V = 0 .
5, and without measurement. The wave packet has initial momentum p = 1, width σ x = 5 , and initial position x = − σ x . Time is measured in unitsof σ x /p . Red values are positive, blue values are negative, and black is zero. Thecorresponding animation is included in the supplementary data. t=2 (a) t=4 t=6 t=8t=2 (b) t=4 t=6 t=8 p -3-2-10123 x -20 -10 0 10 20 t=2 (c) t=4 t=6 t=8 Figure 3.
Wigner functions for a particle incident on a step-function measurementprofile µ ( x ) = θ ( x ) with κ = 5; the profile is the same as the potential in Fig. 2. Thetrajectories are generated with a white-noise-measurement unraveling, as given by theEq. (10). The initial conditions are the same as in Fig. 2. (a) Single trajectory withreflecting measurement outcome. (b) Single trajectory with transmitting measurementoutcome. (c) Ensemble average over measurement realizations. Red values are positive,blue values are negative, and black is zero. The corresponding animations are includedin the supplementary data. eflection of a Particle from a Quantum Measurement x > x > x > κ − . Since the measurement time scale is fasterthan that of the motion in the strong-measurement limit, the particle is localized in thesmall, evanescent tail before it can propagate away. The particle will now have a broad,positive momentum distribution and will propagate away from the step, as shown aftertransmission occurs in Fig. 4. The transmission process is incoherent: the state of theparticle is entangled with the state of the bath, which destroys any interference withthe incident wave packet. Since transmission occurs at random times, the transmittedwave packets will have different phases and propagate away at different times.Fig. 3(c) shows an ensemble average of 10 trajectories for particles incident onthe step measurement. Note that the specular component of the reflection survives theensemble average, while other features related to the measurement noise are washed outin the average. The transmitted parts show no coherence with one another or with thereflected parts. The incoherent parts of both reflected and transmitted wave packetsshow heating from the measurement back-action.Surprisingly, part of the reflection is coherent. It is possible to see interferencebetween the reflected and incident wave packets, even though the reflection is caused bythe back-action of a measurement, which conventionally leads to diffusion or heating.The interference is marked by the alternating positive and negative regions centered at p = 0 in the Wigner function. The interference fringes have the same phase for eachmember of the ensemble—that is, they are the same for each possible, random outcomeof the measurement—producing the coherent reflection.The measurement-induced reflection is analogous to interaction-free measure-ments [44]. In those experiments, light detects the presence of an absorber in the armsof a polarization interferometer without ever interacting with the absorber. An optical eflection of a Particle from a Quantum Measurement x -15 15-10 -5 0 5 10 y o o oo |o o o ( x ) o| before after p -7 7-5 -2.5 0 2.5 5 f o o oo |o o o ( p ) o| (cid:9) beforeafter -7 7010 -4 before after Figure 4.
Position and momentum distributions for a particle before and after ittransmits through the step measurement. The initial conditions are the same as inFig. 2. The time elapsed between “before” and “after” is ∆ t = 0 . κ − . Note that thisis not the same trajectory as in Fig. 3(b), but it displays a similar behavior. Inset: logplot of the same momentum distribution. example of an interaction-free measurement closely related to the reflection phenomenonhere is that of a high-finesse Fabry–Perot cavity [45, 46]. An empty cavity transmitslight almost perfectly, but when an absorber is placed in the cavity to “detect” the light,a large fraction of the light instead reflects from the cavity, although the reflected parthas not interacted with the absorber. We now consider the case of a Gaussian measurement function. This is an even functionof position, so we expect this measurement to tend to drive the particle towards spatialsuperposition states. In addition, if we consider the resonance-fluorescence scenario,a laser probe with a Gaussian profile would realize this Gaussian measurement. Themeasurement function is µ ( x ) = exp (cid:20) − x σ µ (cid:21) , (25) eflection of a Particle from a Quantum Measurement σ µ sets the width of the measurement. Note that we take µ ( x ) to be dimensionless,which is important to keep in mind when comparing to standard measurements µ ( x ) = x , which makes sense only when x is taken to be dimensionless.Fig. 5 shows a wave packet incident on a Gaussian potential barrier without anymeasurement. The wave packet has enough potential energy to cross the barrier (withouttunneling). As with the step potential, the wave packet coherently splits into reflectedand transmitted components. Note that in this case, both parts of the wave packet havethe same asymptotic momentum after scattering from the potential barrier.Fig. 6(a) shows a single trajectory where the majority of the wave packet reflectsfrom the Gaussian measurement, without an explicit potential. Since the Gaussian is asmooth function, the wave packet readily enters regions where the measurement strengthis small. The Gaussian function lacks a sharp edge, and it is possible to get a seriesof results that weakly localize the atom in the soft edges of the function. A small parttransmits through on this particular trajectory, as indicated by the striations showingthe coherence between the reflected and transmitted parts of the wave packet.Fig. 6(b) shows an alternative case where the wave packet is localized by theGaussian measurement. The sides of the Gaussian act as a linear measurement forwave packets confined to one side. Each side can distinguish different positions withinthat side and act to localize the wave packet inside the measurement region. The wavepacket then diffuses around from the momentum kicks of the measurement. This hasoccurred by t = 4 in Fig. 6(b). If the wave packet passes over the peak of the Gaussianthen it will be driven to a superposition state, since the measurement is locally evenin x at the maximum. If the wave packet is moving quickly then the superpositionwill tend to be biased in the direction of motion. The creation of a superposition stateis probabilistic, and depends both on localizing the wave packet, and the wave packetdiffusing over the peak of the measurement function. After the superposition forms, thetwo components diffuse away from the peak of µ ( x ). The symmetry of the measurementensures that the superposition is not disturbed once it is created. A superposition statehas been created by t = 6 in Fig. 6(b).Fig. 6(c) shows the ensemble average over 10 trajectories for a Gaussianmeasurement. Once again, the coherent reflection survives the ensemble average. Theinterference fringes between the reflected and incident wave packets are easier to see,since the incoherent features from weak localization and partial transmission haveaveraged out. Both the reflected and transmitted parts of the wave packet have anincoherent, heated part; this comes from realizations where the particle is localized andsplit into a superposition state. In general, the phase of each superposition is randomand the superpositions are created at random times, so the coherent features of eachsuperposition are washed out in the ensemble average. The ensemble-averaged resultsdo not condition on any measurement scheme, and correspond to the unconditioneddensity operator—the solution to the unconditioned master equation (13). Thus wewill see coherent reflection from any process that realizes master equation (13) in theensemble average. eflection of a Particle from a Quantum Measurement p -4-3-2-101234 x -20 -10 0 10 20 t=2 t=4 t=6 t=8 V ( x ) / V Figure 5.
Wigner functions for particle incident on a static Gaussian potential, withpotential width σ µ = 1 and height V = 0 .
25, without measurement. The wave packethas initial momentum p = 1, width σ x = 5 , initial position x = − σ x + σ µ ). Timeis measured in units of ( σ x + σ µ ) /p . Red values are positive, blue values are negative,and black is zero. The corresponding animation is included in the supplementary data. t=2 (a) t=4 t=6 t=8t=2 (b) t=4 t=6 t=8 p -4-3-2-101234 x -20 -10 0 10 20 t=2 (c) t=4 t=6 t=8 Figure 6.
Wigner functions for particle incident on a Gaussian measurement withmeasurement width σ µ = 1 and strength κ = 5; the profile is the same as the potentialin Fig. 5. The initial conditions are the same as in Fig. 5. The trajectories aregenerated with a white-noise-measurement unraveling, as given by the Eq. (10). (a)Single trajectory with reflection from measurement. (b) Single trajectory producinga spatial superposition state. (c) Ensemble average of measurement. Red values arepositive, blue values are negative, and black is zero. The corresponding animations areincluded in the supplementary data. eflection of a Particle from a Quantum Measurement We can estimate the momentum transferred by the measurement by considering theFourier transform of the measurement function. This is related to the transversemomentum distribution of the bath particles that interact with the particle undergoingthe position measurement. In the resonance-fluorescence example, the bath particlesare photons that form the mode profile of the intensity distribution which becomesthe measurement function µ ( x ). We obtain the rate of momentum disturbance bymultiplying the Fourier transform of µ ( x ) by the measurement rate κ , which is therate at which we gather position information—and hence disturb momentum. Theapproximate final momentum distribution will be the convolution of the particle’smomentum distribution with the Fourier transform of the measurement function.For example, the Fourier transform of the step function θ ( x ) is (cid:90) dk θ ( x ) e ikx = πδ ( k ) + i P . V .k , (26)where P . V . denotes the Cauchy principal value. The sharp edge of the step impliesan arbitrarily fine resolution of how close the particle is to the edge, which impliesa divergent momentum disturbance. The convolution with 1 /k leads to long tails inmomentum space and a divergent momentum uncertainty.The convolution of momentum distributions as a result of “which-way”measurements has been noted before in the context of projective measurements [47]and transverse momentum transfer. The convolution result is exact for a projectivemeasurement, but must incorporate dynamical evolution for continuous measurement.The Fourier transform of a Gaussian is also Gaussian, so the momentum transferin this case is finite. The width of the momentum distribution scales as the inverse ofthe width of the measurement resolution. A narrow measurement function constitutesa locally stronger measurement of position and hence a larger disturbance.There is an alternative, measurement-theoretic explanation for the heating. Astrong measurement ensures that we detect if the particle crosses between regions ofdiffering µ ( x ). Consider the time when the particle begins to cross into the next region,as in the t = 2 and t = 4 panels of Fig. 3(b), or the t = 2 panel of Fig. 6(b). If themeasurement record does not yet indicate that the particle has crossed through, thenthe particle can only have penetrated a small distance into the next region. Then, whenthe measurement indicates that the particle has passed into the next region, the statecollapses down to the small part that penetrated the next region. Since this part istightly confined in position, there now is a large uncertainty in momentum. The wavefunction is effectively multiplied by µ ( x ), and so the wave function will acquire thesame character in momentum space. This is apparent, for example, in the t = 4 frameof Fig. 6(b). eflection of a Particle from a Quantum Measurement k r e f l ec t i o n p r o b a b ili t y o=o1 p o=o3p o=o5 Figure 7.
Measurement-induced reflection probabilities for a wave packet with σ x =10 incident on µ ( x ) = θ ( x ). Numerical simulations (averaged over 256 trajectories) arecompared with the analytical result from Eq. (32).
4. Quantum Zeno Effect
Despite the heating, a particle will almost certainly reflect from a strong measurement.As Fig. 7 shows, the probability for reflection tends to unity as the measurement strengthincreases. This is a manifestation of the quantum Zeno effect [21–24, 48, 44, 49].A simple argument in the case of a step-function measurement µ ( x ) = θ ( x ), asin Sec. 3.1, demonstrating this in the limit of a sequence of projective measurementsseparated by infinitesimal time intervals ( κ −→ ∞ ) is as follows [29]. We assume thatat time t the particle is completely confined to the region x <
0, as enforced by theprojective measurement in the immediate past. The operator for evolution betweenmeasurements is U ( dt ) = exp (cid:18) − i H (cid:126) dt (cid:19) = 1 − i H (cid:126) dt. (27)The Hamiltonian is responsible for any evolution that could transfer the wave packetinto the region x >
0. The term proportional to dt is the probability amplitude forthe particle to cross into the region x > x < x > O ( dt ) = 0 for the next projective measurement,resetting the particle to the x < eflection of a Particle from a Quantum Measurement µ ( x ) = θ ( x ), d | ψ (cid:105) = − i p m (cid:126) | ψ (cid:105) dt − κ (cid:0) θ ( x ) − (cid:104) θ ( x ) (cid:105) (cid:1) | ψ (cid:105) dt + (cid:18) θ ( x ) | ψ (cid:105)(cid:104) ψ | θ ( x ) | ψ (cid:105) / − | ψ (cid:105) (cid:19) dN, (28)where dN is a Poisson process with mean (cid:104)(cid:104) dN (cid:105)(cid:105) = 2 κ (cid:104) θ ( x ) (cid:105) dt. (29)This stochastic Schr¨odinger equation gives the same unconditioned evolution as thewhite-noise unraveling in Eq. (10). We will model the incoming wave packet as aunit-amplitude plane wave, which gives the correct reflection coefficient as long as themomentum distribution is narrow compared to the mean incident momentum.We can use a linear trajectory and solve for the between-jump evolution (thenonlinear term represents the probability of the particle to be detected in the region x >
0, which is precisely what we are computing here). The linear, between-jumpevolution is governed by i (cid:126) ∂ t | ψ (cid:105) = (cid:20) p m − i (cid:126) κθ ( x ) (cid:21) | ψ (cid:105) . (30)We assume the particle is initially located on the left with mean energy p / m , andenforce continuity of the wave to match the incoming wave with a reflected wave of thesame energy and a decaying solution for x >
0, which gives the total probability for theparticle to be in the “transmitted” region. This is used along with the mean detectionrate to find the total detection probability: P det = 2 κ ∞ (cid:90) −∞ dx θ ( x ) | ψ ( x ) | (31)= 2 √ ξ/χ (cid:0) √ ξ /χ (cid:1) + χ , (32)where ξ := 2 m (cid:126) κ/p is the measurement strength scaled in units of the incident kineticenergy, and χ := (cid:113)(cid:112) ξ −
1. For large κ the detection probability, and thus thetransmission probability, vanishes as κ − / .In this linear-trajectory picture, the measurement can be viewed as an imaginarypotential that absorbs particles. The absorbed particles are detected inside the x > | ψ (cid:105) → θ ( x ) | ψ (cid:105)(cid:104) ψ | θ ( x ) | ψ (cid:105) / . (33)The remainder of the particles coherently reflect from the measurement with aprobability given by the norm of the state.In Fig. 7 we have counted only the coherent part of the reflection by computing +3 σ p (cid:82) − σ p dp | φ ( − p + p ) | , where φ ( p ) is the momentum-space wave function. This omits the eflection of a Particle from a Quantum Measurement x > κ increases. As κ increases the probability for reflection increases as the wavepacket is excluded from the measurement region. Concomitantly, any increase in κ increases the heating from the measurement since the evanescent tail in the absorbingregion is more tightly confined. This offsets the decrease in mean momentum from thereflection.The Zeno effect has also been noted when combining a continuous positionmeasurement of µ ( x ) = x with a double-well potential [6]. For moderate measurementstrengths, the position measurement localizes the particle in one well and inhibits thetunneling between the two wells. This breaks down for large measurement strengthssince the heating from the measurement boosts the energy of particle above the barrierin the middle. In our case the Zeno effect arises because of the inhomogeneity of themeasurement function, rather than the inhomogeneity of the potential.
5. Physical Realizations
The above effects can be realized by coupling a two-level atom to a high-intensitylight field with position-dependent Rabi frequency. This can be viewed in terms ofthe stochastic dipole force [41]. We write the position-dependent Rabi frequency asΩ( x ) = | Ω | g ( x ), where | Ω | = | d eg · E | / (cid:126) , | E | is the maximum electric field amplitude,and g ( x ) is a complex, dimensionless mode profile with | g ( x ) | ≤
1. The atom obeys ∂ t ρ = − i (cid:20) p m (cid:126) + ∆ σ † σ + 12 (cid:2) Ω ∗ ( x ) σ + Ω( x ) σ † (cid:3) , ρ (cid:21) + Γ D [ σ ] ρ, (34)where Γ is the excited state decay rate. In the high intensity limit, the dressed states |±(cid:105) are approximately equal superpositions of the ground state | g (cid:105) and the excited state | e (cid:105) , and vice versa. Each spontaneous-emission event projects the atom into the groundstate, and thus into an equal superposition of the dressed states. The atom thereforesees both shifts ± (cid:126) | Ω( x ) | / |±(cid:105) . The interpretation is eflection of a Particle from a Quantum Measurement F = ∓ (cid:126) ∇| Ω | , (35)where the sign is chosen randomly, but with equal probability for the two possibilities.Assuming the atom is moving slowly, the momentum change associated with a singlespontaneous-emission event is∆ p = ∓ (cid:126) ∇| Ω | ξ (36)where ξ is the time until the next spontaneous emission event, which is a random variable( ξ >
0) of mean 2 / Γ and exponential probability density f ( ξ ) = Γ2 exp (cid:18) − Γ2 ξ (cid:19) . (37)To take into account the randomness of the sign, we can write∆ p = (cid:126) ∇| Ω | ξ (cid:48) (38)where ξ (cid:48) ∈ R has a two-sided exponential probability density f ± ( ξ (cid:48) ) = Γ4 exp (cid:18) − Γ2 | ξ (cid:48) | (cid:19) . (39)Then the mean-square kick is (cid:104) (∆ p ) (cid:105) = (cid:126) ( ∇| Ω | ) (cid:104) ξ (cid:48) (cid:105) = 2 (cid:126) ( ∇| Ω | ) Γ , (40)where (cid:104) ξ (cid:48) (cid:105) = Γ4 (cid:90) ∞−∞ dξ (cid:48) exp (cid:18) − Γ2 | ξ (cid:48) | (cid:19) ξ (cid:48) = 8Γ . (41)The diffusion rate is the mean-square step divided by the average step time ∆ t = 2 / Γ,so D p = (cid:104) (∆ p ) (cid:105) ∆ t = (cid:126) ( ∇| Ω | ) Γ = (cid:126) | Ω | Γ | g (cid:48) ( x ) | . (42)This heuristic result agrees with the high-intensity limit of more rigorous butcumbersome calculations [55, 41, 56].Since the mean time Γ / V ( x, t ) = | Ω |√ Γ (cid:112) | g ( x ) | ◦ dWdt , (43)where we have dropped the mean dipole potential, which vanishes on resonance ∆ = 0.We can see that this form of the potential is correct by comparing Eqs. (14) and (15), eflection of a Particle from a Quantum Measurement κ = | Ω | /
2Γ and µ ( x ) = (cid:112) g † ( x ) g ( x ), where the sign of µ ( x ) is defined such that | µ (cid:48) ( x ) | = | g (cid:48) ( x ) | . Wecan then read off the momentum-diffusion coefficient using Eq. (23) to see that it agreesexactly with Eq. (42).It is interesting to note that in this high-intensity regime, the stochastic potentialrepresents an interplay between the fluctuations in the atomic dipole and the fieldintensity. That is, this effect is due to fluctuations in the atom–field interactionHamiltonian itself. As such, it does not saturate for high intensities, as does spontaneousemission.A measurement interpretation of the stochastic dipole force is as follows. Thedipole force for an atom in a particular dressed state arises due to coherent scattering ofphotons between different wave vectors of the driving field. In principle, this coherentredistribution of light can be measured to obtain information about the position of theatom in the field, as for example is the case in recoil-induced resonances [57]. However,on resonance, the atom flips randomly between the dressed states, so that the mean redistribution of photons (and hence the mean dipole force) vanishes. The measurementinformation is instead encoded in the variance of the redistributed light.Note that a low-intensity jump decomposition of resonance fluorescence has beenconsidered in the context of realizing imaginary potentials [58], though without exploringthe consequences for the atomic dynamics. Unfortunately, in the low-intensity limit thespontaneous emission recoil obscures the measurement effects, and this model can breakdown when pushed to the high intensities required for a large measurement strength. A spatially varying position measurement can also be realized in the interaction of atwo-level atom with the field of an optical cavity, which is in turn driven by a classicalfield. Due to a large detuning, and a large cavity decay rate we can eliminate theatom’s internal dynamics and the cavity dynamics, leaving the only the center-of-massmotion of the atom. In this case the cavity mode function becomes the measurementfunction [9, 13, 14]. The simulations from Sec. 3.2 correspond to Gaussian mode of aring cavity.We assume that the cavity is being monitored by homodyne detection, so the systemobeys the following master equation [14] dρ = − i (cid:126) [ H, ρ ] dt + Γ D [ σ ] ρ dt + γ D [ a ] ρ dt + √ γ H [ a ] ρ dW, (44)where Γ is the free-space spontaneous emission rate, γ is the cavity decay rate, and a isthe cavity annihilation operator. The total Hamiltonian is given by H = p m + (cid:126) ω C a † a + (cid:126) ω A σ † σ + (cid:126) g ( x )( a † σ + aσ † )+ (cid:126) E ( ae iω C t + a † e − iω C t ) , (45)where ω C is the cavity mode resonance, ω A is the atomic transition frequency, g ( x ) isthe cavity mode profile, and E = (cid:112) γP/ (cid:126) ω C is the amplitude for the classical driving eflection of a Particle from a Quantum Measurement P is the power in the driving field. If we transform to a rotating frame,we can remove the free evolution due to the cavity. Since the classical driving field isresonant with the cavity, we can then write H = p m + (cid:126) ∆ σ † σ + (cid:126) g ( x )( a † σ + aσ † ) + (cid:126) E ( a + a † ) , (46)where ∆ = ω A − ω C . In the limit ∆ (cid:29) g ( x ) , γ, Γ , p / (2 m (cid:126) ), we can adiabaticallyeliminate the excited state. This amounts to replacing σ and σ † with (cid:104) σ (cid:105) and (cid:104) σ † (cid:105) . (Fora more rigorous approach see Ref. [9].) In this limit the effective Hamiltonian is givenby H eff = p m + (cid:126) g ( x )∆ a † a + (cid:126) E ( a + a † ) , (47)and the spontaneous emission terms can be dropped since the effective decay rate is O [∆ − ]. If we further assume that the cavity is strongly damped such that γ (cid:29) g / ∆ , E, p / (2 m (cid:126) ), we can also eliminate the cavity. The final result is the followingmaster equation for the center-of-mass dynamics, dρ = − i (cid:126) [ H eff , ρ ] dt + 2 κ D [ µ ( x )] ρ dt + √ κ H [ µ ( x )] dW, (48)where H eff = p m + (cid:126) α g ∆ µ ( x ) κ = α g ∆ γµ ( x ) = g ( x ) g , (49)where α = 2 E/γ , and g = max[ g ( x )]. The mean potential in the effective Hamiltoniancan be cancelled by the Stark shift from an off-resonant classical field that does notresonate with a cavity. This interaction has the form H Stark = − (cid:126) | Ω( x ) | δ . (50)Cancelling the mean potential allows the measurement effects we have discussed to cometo the fore. The master equation then has the same form as Eq. (10). Measurements can also be viewed as imaginary (absorbing) potentials, as seen in Sec. 4.Imaginary potentials have been used to coherently diffract atomic beams [59, 60] andreshape wave functions to counteract the expansion of a free particle [61]. A resonantstanding wave of light acts like an imaginary potential for two-level atoms. When theatom absorbs a photon from the resonant standing wave, it will decay to another state,thus being effectively absorbed from the initial beam. The diffraction and reshapingboth rely on post-selecting atoms that have not undergone spontaneous emission. Thesurviving atoms are thus most likely to be near the nodes of the standing wave. For eflection of a Particle from a Quantum Measurement ( kx ) measurement of position,while the diffraction is a consequence of scattering off the periodic array of nodes.
6. Analogies
The coherent reflection from a measurement is analogous to the coherent back-scatterof light from a disordered atomic medium [64]. The analogy is clearer from thefluctuating-potential picture, since the paraxial wave equation is the same as theSchr¨odinger equation with time replaced by the propagation direction. Under the samecorrespondence, the fluctuating potential in time becomes a disordered potential inspace. In the ensemble average the fluctuating potential also yields a coherent reflection,since it obeys the same unconditioned equations as the measurement.An alternative analogy is the reflection of a photon from a conducting surface. Thelarge complex permittivity—whose imaginary part represents absorption—leads to arapid extinction of the wave inside the conductor if the conductivity is large. This isanalogous to a large measurement strength for detecting the electromagnetic wave insidethe region. This leads to the large reflection probability for light from the surface of agood conductor, even though it is a good absorber for waves inside the medium.The above examples capture the coherence of the reflection, but miss thetransmission aspects of the measurement since the inbound particle is absorbed by theinteraction. Unlike photons, atoms are not destroyed upon detection, so the form of thetransmitted wave packet is necessary for a complete picture of measurement-inducedreflection.
7. Outlook
We have outlined a theory of continuous position measurements that describes a spatiallyvarying measurement strength. The effects shown here require the measurement gradientto dominate over all other dynamics. It is only in the limit of a large measurementgradient on the scale of the atom, large intensities and small velocities that coherentreflection can take place. This work can be extended to include other couplings andposition measurements such as atoms coupled to micro-toroidal resonators [65], and theeffect of measurements when the particle is confined to a potential. The case of both aninhomogeneous measurement and potential is of interest, for example, for studying thequantum–classical transition for classically chaotic systems. Future work will extendthe idealized theory presented here to account for the imaging setup, diffraction effectsand realistic photo-detection.
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8. Acknowledgments
The authors wish to acknowledge discussions with Tanmoy Bhattacharya, Robin Blume-Kohout, Steven van Enk, and Michael Raymer, and critical readings of the manuscriptby Eryn Cook, Paul Martin, Elizabeth Schoene, and Jeremy Thorn. JBM and DAS aresupported by the National Science Foundation, under Project No. PHY-0547926. KJis supported by the National Science Foundation under Project No. PHY-0902906.
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