Imaginary past of a quantum particle moving on imaginary time

The analytical continuation of classical equations of motion to complex times suggests that a tunnelling particle spends in the barrier an imaginary duration $i|\mathcal T|$. Does this mean that it takes a finite time to tunnel, or should tunnelling be seen as an instantaneous process? It is well known that examination of the adiabatic limit in a small additional AC field points towards $|\mathcal T|$ being the time it takes to traverse the barrier. However, this is only half the story. We probe the transmitted particle's history, and find that it"remembers"very little of the field's past behaviour, as if the transit time were close to zero. The ensuing contradiction suggests that the question is ill-posed, and we explain why.

The analytical continuation of classical equations of motion to complex times suggests that a tunnelling particle spends in the barrier an imaginary duration i|T |.Does this mean that it takes a finite time to tunnel, or should tunnelling be seen as an instantaneous process?It is well known that examination of the adiabatic limit in a small additional AC field points towards |T | being the time it takes to traverse the barrier.However, this is only half the story.We probe the transmitted particle's history, and find that it remembers very little of the field's past behaviour, as if the transit time were close to zero.The ensuing contradiction suggests that the question is ill-posed, and we explain why.

I. INTRODUCTION
Recent advances in atto-second laser techniques [1] have revived the interest in the nearly century-old question [2].Is tunnelling infinitely fast as was suggested, for example, in references [3], [4], or does a particle spend in the barrier a finite duration, as was argued, e.g., in [5], [6]?The question itself is somewhat vague, since there is no consensus as to how this duration should be defined (for recent and not-so-recent reviews of the subject see [7]- [13]).
In dealing with a conceptual difficulty it is often helpful to resort to a simple yet representative model.One such model is that of a semiclassical particle transmitted across a smooth potential barrier.It is well known that the particle's motion can be described by the equations of classical mechanics analytically continued into the complex time plane [14], [15].This provides a formal yet not a particularly useful answer: The particle spends in a classically forbidden region an imaginary duration i|T |.The result clearly needs an interpretation, and in Ref. [16] the authors, who studied the adiabatic limit of tunnelling in the presence of a small AC field, concluded that |T | must after all be the time the particle spends in the barrier region [17].In this paper we show that analysis of [16], [17] (the model was recently revisited in [18]) presents only one side of the story.Examining instead the part of the time dependent potential actually experienced by a tunnelling particle equally suggests that the duration spent inside the barrier must in fact be close to zero.We argue that the question is ill-posed, and explain why it should be so.Before addressing the case of tunnelling we briefly review the classically allowed case, where both approaches agree.

II. THE CLASSICALLY ALLOWED CASE
Consider a particle of energy E and mass m, incident from the left on a stationary potential barrier V (x), as-suming E > V (x).A small time dependent perturbation [θ ab (x) = 1 for a ≤ x ≤ b, and 0 otherwise] is added in a region a < x < b (see Fig. 1).The particle's wave function ψ(x, t) satisfies a Schrödinger equation (ℏ = 1) which can be solved by expanding the wave function in , where where K(x, y, t − τ ) is the Feynman propagator [19] for the motion in a potential V (x).Evidently, ψ(x, t) may depend on (i.e., "remember") all the values of W (y, τ ), τ ≤ t, although its behaviour in the distant past is expected to be less important.Defining an effective region of integration for an oscillatory integral is notoriously difficult (for more details see, e.g., [20]).However, a more precise estimate is available if V (x) varies slowly compared to the particle's de-Broglie wavelength.Then the terms in the perturbation series can be obtained by invoking the semiclassical approximation for K(x, y, t−τ ) [19], and evaluating the τ -integrals by the stationary phase method.If, in addition, W (x, t) in Eq.( 2) also varies slowly in x (except maybe at the endpoints x = a and x = b), the series can be summed (for details see Appendix A).Thus, for an x > b, "downstream" from the region which contains w, one finds ψ(x, t) ≈ p(x, E) −1/2 exp{i[S 0 (x, t) + S 1 (x, t)]}, (4a) and is the particle's momentum.In Eq. (4c), x(τ ) = x(τ |x, t, E) is the classical trajectory of a particle with energy E, which arrives in x at a time t, implicitly defined by and τ a ≡ τ (a, x, t), τ b ≡ τ (b, x, t) are the moments when the trajectory enters and leaves the region [a, b], respectively.Equations ( 4)-( 6), whose validity is discussed in Ref. [21] and in Appendix B, have a simple interpretation.We are interested in a particle of energy E found ("postselected") in a given location x at a time t.In the (semi)classical limit, the particle's past is well defined: it has been following a classical trajectory x(τ |x, t, E).Accordingly, the particle may probe the potential W only at the moment x(τ |x, t, E) passes through that location.Clearly it spends inside the region [a, b] a duration It is easy to devise simple tests which would convince one that this is, indeed, the case.Consider the case of a time dependent potential "step" and ask how slowly Ω must vary for the particle to "see" a static potential, fixed, say, at the moment it leaves the region [a, b].Choosing x = b, τ b = t helps discount the time of travel from b to x, and expanding Ω(τ ) in a Taylor series, Ω(τ ) ≈ Ω(τ b ) + Ω(τ b )(τ − τ b ), one finds Figure 2. The classically allowed case: A particle with energy E arriving in x at time t probes the values of W (x(τ ), τ ) along its trajectory x(τ |x, t, E) (dashed green) defined by Eq.( 6).
Only two pulses, n = 0 and n = −1 (in orange), contribute to the phase of ψ(x, t) (see Table I).
The first term is the desired adiabatic approximation.
The second one can be neglected if it is small compared to unity.For a harmonic perturbation Ω(t) = cos(ωt) this means ωT ab ≪ 1/w 0 T ab , where As expected, adiabaticity is achieved provided the particle crosses the region [a, b] so quickly, the oscillating potential has no time to change.
Our second test is even simpler.Suppose Ω(t) in Eq. ( 8) is a sequence of Gaussian pulses of various magnitudes provided by the experimenter.How many of these pulses affect the phase of the wave function ψ(x, t) at some (x > b, t)?The answer is, of course, those which overlap with the interval τ a ≤ τ ≤ τ b , since Figure 2 shows the case where the particle leaves the region [a, b] at τ b = 0, T ab is equal to the separation between the pulses, T , and ∆t = T /3.Their relative contributions |S 1,n (x, t)|/|S 1,0 (x, t)| are given in Table I for future comparison with the classically forbidden case.
The same conclusion can also be drawn from inspecting directly observable quantities.The simplest choice would be the probability density ρ(x, t) ≡ |ψ(x, t)| 2 , but it remains unchanged by the presence of W (x, t).The probability current at (x, t), j( may, however, be affected since W (x, t) can alter the particle's velocity.Thus, for the extra term added to is the additional force acting on the particle, the last integral is the work done to change its kinetic energy.The first term in brackets accounts for the jolts experienced by the particle as it enters and leaves the region, and is the only remaining contribution in the case of a step-like potential (8).The current at (x, t) depends on Ω(τ ) only for τ a < τ < τ b , which provides yet another proof of the particle's presence between x = a and x = b during this interval.Note that there is no extra current in the adiabatic limit Ω → 0, since for a static potential the net work done by the extra force acting on the particle is null.However, for W in Eq. (8), δj(x, t) approaches its zero limit as ∼ ΩT ab , and the classical duration spent in [a, b] is again the relevant time parameter.
In summary, both tests are consistent with a classical picture of a particle which enters the region at τ = τ a , leaves at τ = τ b , and spends there a duration τ b − τ a .
The tests can also be applied in the classically forbidden case, and we will do it next.

III. THE CLASSICALLY FORBIDDEN CASE (TUNNELLING)
The formal derivation in the case E < max [V (x)] is remarkably similar, if one extends the definition of the particle's momentum p(x, E) to the region where E < V (x) as ( It is easy to check (see Appendices A and B), that the wave function of a particle transmitted across the barrier is still given by Eq.( 4) where and x < and x > are the turning points where p(x, E) vanishes (see Fig. 1).The main difference with the classically allowed case is that now integration over τ is performed along a contour Γ in the complex τ -plane, defined by, The contour, shown in Fig. 3, runs along the real τ -axis while x(τ ) ≥ x > , then parallel to the imaginary axis for x between x < and x > , and finally parallel to the real axis for x < x < .It is readily seen that on Γ Eq. ( 15) defines a real-valued trajectory x(τ |x, t, E) (Im[x] = 0), shown in Fig. 3 for an Eckart potential [22].The trajectory satisfies the Newton's equation m ẍ(τ ) = −V ′ [x(τ )] along each segment of the contour, and was proven by McLaughlin [15] to be the real-valued critical path of the Feynman path integral analytically continued to complex times.The time integrals in the power series for ψ(x > b, t) can now be evaluated by the saddle point method (see Appendix A), and the summed series bears similarity to that in Eq.( 4).There is, however, an apparent difficulty in trying to establish when exactly the tunnelling particle was present in the classically forbidden region, since a perturbing potential (8), restricted to the sub-barrier region, a = x < , b = x > , is probed along the vertical part of the contour Γ in Fig. 4. For a particle, just emerging from the barrier at x = x > , t = τ b = 0, τ a = i|T x < x > |, the integration is along the imaginary time axis, and one has An experimenter, controlling the real-time behaviour of the perturbing potential given by (8), is unlikely to be satisfied with an explanation that "the particle is present inside the barrier between the times τ a = i|T x < x > | and τ b = t", and would want to know what this could mean in practice.In case of tunnelling, he/she will need to look no further than the particle density at (x, t) now modified by the perturbation, since S 1 (x, t) is complex valued.
As in our first adiabatic test, one can ask how slowly should Ω(τ ) vary on the real τ -axis, for ρ(x, t) to be given by its expression for a static step-like potential w 0 Ω, in which constant Ω is replaced by Ω(t) for each t.Expanding, as before, Ω(t + iz) in a Taylor series around τ b yields Now with Ω(t) = cos(ωt) the adiabatic limit ρ(x, t) = ρ 0 (x) exp −2w 0 cos(ωt)|T x < x > | is reached provided the first and the second terms on the right-hand side of Eq. ( 18) are of order of unity, and much less than unity, respectively.This leads to a condition first obtained in Ref. [16], albeit in a slightly different form.Its similarity to the classically allowed case convinced the authors of Refs.[16], [17] that |T x < x > | must be "the time a tunnelling particle spends in the barrier region".We will, however, reserve judgement until submitting the tunnelling particle to our second test.Suppose again that the potential in the classically forbidden region is modified by a series of Gaussian pulses (10).If |T x < x > | is indeed the duration spent in the barrier region, the particle exiting the barrier at x = x > and t = 0 must be affected by what happened in the barrier between t = −|T x < x > | and t = 0. This, however, does not happen, as we will show next.Calculating ρ(x, t) at x = x > and t = 0 with the help of Eqs.( 16) and ( 17) yields Unlike in the classical case, increasing |T x < x > | (e.g., by making the barrier broader) does not let the particle experience more pulses, since Im [S 1,n ] /Im [S 1,0 ] remains of order of exp − n 2 T 2 ∆t 2 , small if ∆t ≪ T .Moreover, the approximation used in deriving Eqs. ( 16) and ( 17) requires S 1,n ≲ 1, so that, by necessity, Im [S 1,n̸ =0 ] ≪ 1, which leaves S 1,0 the only non-negligible contribution.The particle appears to have "no memory" of other narrow pulses, and the experimenter would measure practically the same density ρ(x > , 0) by applying only the Ω n=0 (τ ) pulse.The ratios are listed in Table I alongside the results obtained earlier for the classically allowed case.With our choice of parameters, in the classical regime the pulses with n = 0 and n = −1 contribute equally to S 1 in the exponent in Eq.(4a).With tunnelling, only the n = 0 pulse affects the particle emerging from the barrier, which should not be the case for a particle spending |T x < x > | in the subbarrier region.
There is obviously a discrepancy with what was learnt in the adiabatic test, and it is easy to see why.In our first test, the rate of change of a slowly varying analytical function Ω(τ ) along any direction in the complex τ -plane is determined by the same Ω(τ b ), which accounts for the similarity between Eqs.( 9) and (18).However, in the second test, analytic continuation of a Gaussian is such that making barrier broader does not let the particle explore its behaviour further into the past.(The reader may note a certain parallel with the Hartman effect [23], whereby the arrival of a tunnelled particle is not delayed by increasing the barrier's width.)The effect persists if the ratio ∆t/|T x < x > | is made smaller (see Appendix B), which suggests that the duration the particle spends in the barrier must be zero, or very close to zero.One can keep the ratio ∆t/T constant, and reduce T so more pulses fit into the interval − T x < x > , 0 (w 0 can be cho-sen to ensure S 1 ∼ 1.) Still, only the n = 0 pulse will be taken into account, even though now the n = −1 pulse belongs to a more recent past.

IV. CONCLUSIONS AND DISCUSSION
Our aim here is not to provide yet another "evidence" of tunnelling being "instantaneous", but rather to gain a broader view of the problem itself.Two equally feasible (at least in principle) experiments give the same answer in the classically allowed case.Both suggest that the interaction of a classical particle with a small field is governed by the duration T , spent by its trajectory region where the field is applied.Alice and Bob, making different tests in their respective laboratories, can agree about this point.This is not so in the case of tunnelling.An extension of the classical equations of motion to complex time still yields a single "duration" T , but now it is imaginary, T = i|T |.Is tunnelling instantaneous, since Re[T ] = 0, or does it take |T | seconds to cross the barrier region?Both tests can be performed also in the tunnelling regime, but this time the results appear to contradict each other.With the classical picture still in mind, Alice's would find her adiabatic test pointing towards a finite tunnelling time, |T |.This is the well-known Büttiker-Landauer result, often interpreted as the true duration of a tunnelling process [16].On the other hand Bob, who studies the "memory" of a tunnelled particle, must conclude that the duration spent in the barrier is much shorter than |T |, and is in fact close to zero.If both Alice's in Bob's results are correct, the concept of a well-defined classicallike tunnelling time must be at fault.The problem is well known in quantum measurement theory.A quantity, uniquely defined on a single (classical) trajectory must become indeterminate if two or more paths leading to the particle's final condition interfere [33].An attempt to measure it without destroying the interference yields only a complex "weak value" (see, e.g., Ref. [24] and references therein).Such is the time measured by a weakly coupled Larmor clock [25]- [30], The spin of the clock results rotated both in the plane perpendicular to the magnetic field by an angle φ 1 ∼ τ 1 (as in the classical case), but also in the plane parallel to the field, by φ 2 ∼ τ 2 (a new nonclassical feature).Moreover, in the semiclassical limit one has τ 1 ≈ 0, whereas τ 2 ≈ |T | (the reader may begin to see a connection).Alice, who has measured φ 1 ≈ 0, could argue that "tunnelling is instantaneous", since the spin has not rotated in the way it does in the classical case.Bob, who has measured φ 2 ≈ |T |, could argue that tunnelling "takes a finite time" since, after all, the field caused the spin rotate, albeit in a different manner.So, which part of the complex τ L represents the physical "duration spent in the barrier"?
In general, neither τ 1 nor τ 2 is a valid candidate since, for some initial and final particle's states, both can turn out to be negative [31], [32].The absolute value, |τ L | (known as the "Büttiker time" [26]) is indeed real, but can exceed the time the particle was in motion [31], [32].The reason for this strange behaviour is the Uncertainty Principle [33] which, among other things, forbids knowing the time spent in the barrier in a situation where different durations interfere to produce the tunnelling amplitude [32].
If the theory does not allow one to describe classically forbidden tunnelling in terms of a classical-like duration, it is rather pointless to ask whether or not this nonexistent duration should be zero.One can try to approach the question from the operational side, by adopting to the tunnelling regime experiments, known to yield the classical duration in the classically allowed case.The problem is, one will obtain results which, if interpreted at face value, appear to support either of the conflicting viewpoints.In this paper we have shown that the same is also true for the Büttiker-Landauer model [16], which employs a time-modulated barrier, instead of an external clock.A reasonable way out of the contradiction is to accept that, in the absence of a classical-like tunnelling time, different experiments may yield different quantities with units of time [34].The nature and properties of such quantities can then be analysed individually by the methods of elementary quantum mechanics.
One may be disappointed to learn that the object of a long standing search may not exist or, what is worse, exists in many conflicting shapes and forms.Perhaps a different approach would deliver a well defined tunnelling time?This seems unlikely.The "phase time" (see, e.g., Ref. [7]), associated with the transmission of broad wave packets, can be analysed along the same lines as the Larmor time [20], [35], [36].Moreover, the above analysis emphasises the importance of a conventional trajectory for the existence of a unique duration spent in a given region of space.Even when such a trajectory exists along a contour displaced into the complex time plane (cf.Fig. 3), the resulting complex duration looses the properties of a physical time interval.We need to solve, in the semiclassical limit, a Schrödinger equation where W (x, t) vanishes unless a ≤ x ≤ b.
As ℏ → 0, the time integral for k = 1 can be evaluated by the stationary phase method.The corresponding condition ∂ τ S(x, y, t − τ ) − E = 0 ensures that the main contribution to the integral comes from a time τ s at which a particle with energy E, destined to arrive at (x, t), passes through the point y, With a pre-exponential factor W (y, τ ) evaluated at τ = τ s , the first term of the series [see Eq.(A2)] takes the form Repeating the calculation, using a relation and summing the series for ψ(x, t) yields where, explicitly, p(y ′ ) yields the last of Eqs.(4).A classical particle "probes" a time dependent perturbation as it follows its trajectory.

The classically forbidden case
The case where a particle tunnels across a barrier V (x) can be treated in a similar manner (at least in one spatial dimension), as was shown by McLaughlin in Ref. [15].After applying the standard connection formulae and multiplying the result by exp(−iπ/4) the zero-order wave function can be written as [15], the semiclassical limit of K(x, y, t − τ ) analytically continued into the complex plane of τ still given by x = x < and x = x > , where conditions (B5) and (B7) are clearly violated.However, a more detailed analysis (see Ref. [21] and references therein) shows that the rules remain unchanged near a linear turning point, V (x) ≈ ∂ x V (x > )(x − x > ), W (x, t) ≈ W (x > , t).
The simplest illustration is given by the case of a broad rectangular barrier, V (x) = V θ ab (x), where the matching rules at x = b depend on the particle's energy E (see Ref. [21]).Condition (B7), applied on both sides of the turning point, eliminates this dependence, leaving ψ(x > b, t) still given by Eq. ( 4), apart from an overall energydependent factor.Suppose, for simplicity, that E = V /2, and Ω(t) = exp −t 2 /∆t 2 .We are interested in the ra-tio between the pulse's width ∆t and the modulus of the complex time, |T (B8) However, |p|(b − a) ≫ 1 is the sub-barrier action, which is always large in the semiclassical limit considered here.This shows that the used approximation ( 4) is valid if the pulse is significantly shorter than the Büttiker-Landauer time |T ab | [16].

Figure 1
Figure1.a) Depending on the energy E, a particle can either tunnel (red, x< and x> are the classical turning points), or pass over the barrier's top (dashed green).b) A timedependent perturbation W (x, t) = w(x)Ω(t) is added between x = a and x = b, a constant w(x) = w0 is shown.

TABLE I .
The ratios |S1,n