The phantom slit effect
James Q. Quach∗ and Maciej Lewenstein
ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Spain
arXiv:1709.09851v1 [quant-ph] 28 Sep 2017
We show that under the weak measurement scheme, the double-slit experiment can produce an
interference pattern even when one of the slits is completely blocked. The initial and final states are
corpuscular, whilst the intermediate states are wave-like, in that it exhibits an interference pattern.
Remarkably, the interference pattern is measured to be vertically polarised, whilst simultaneously
the individual photons are measured to be horizontally polarised. We call this the phantom slit
effect. The phantom slit is the dual of the quantum Cheshire cat.
The conventional view of measurement in quantum
mechanics is that it is a destructive process that irrevocably projects the system into an eigenstate of the observed variable. Weak measurements provides a formal
non-destructive measurement scheme by weakly coupling
the system to an ancilla, and performing a measurement
(projection) on the ancilla in some appropriate basis [1].
Operationally, the ancilla is a measurement device with
a pointer; the interaction of the system and the ancilla
shifts the pointer state proportional to the magnitude of
the observed variable. As the ancilla interacts only very
weakly with the system, the state can evolve without appreciable disturbance.
Weak values seek to represent observables of intermediate states, as the system evolves from an initial to a final
or post-selection state. Judiciously choosing the postselection state onto which to project the ancilla (and system), have been shown to produce anomalous phenomena that have sparked debate over the interpretation of
weak values [2–9]. Amongst the anomalous phenomena
is the quantum Cheshire cat (QCC) effect [10], where the
position of a photon exists in one arm of an interferometer, whilst its polarisation exists in the other arm; the
whimsical name alludes to Lewis Carroll’s Cheshire cat,
whose grin (polarisation) could exist without its body
(photon). Backdropping the spirited debate are many experiments that have used weak values to measure several
observables in different quantum systems [11–17]; this includes the famous Kocsis et. al experiment [18], which
observed the individual trajectories of photons as they
formed the interference pattern in the double-slit experiment. In this experiment, the photon momentum was
weakly measured, and the post-selection was the photon
position.
Here we show an interesting new effect, where the second slit in a double-slit experiment can still affect an
interference pattern even after its has been blocked or
removed - we dub it the phantom slit.
Weak values. Prior to measurement the initial system
state |ψi i and pointer state |mi i are uncoupled. In the
weak measurement scheme, the interaction Hamiltonian
between the system and pointer is
Ĥint = g(t)Ô P̂ ,
(1)
FIG. 1. The phantom slit effect. Slit B is blocked, so that light
emanates only from slit A. A weak value detector moves up
and down in a line parallel to the slit plane. It measures a position dependent interference pattern given by hσ̂V iw = cos φ ,
revealing the wave-like nature of the intermediate states, between the initial and final states. Although the intermediate
states are wave-like, the initial ψi (x) = hx|ψi i and final states
ψf (x) = hx|ψf i are corpuscular. The interference pattern
is vertically polarised, whilst the photons are horizontally polarised. d is inter-slit spacing, and θ is the angle from the normal of the slit plane. φ = kd sin θ, where k is the wavenumber
of the light source.
which couples the system’s observable Ô to the pointer
momentum P̂ . The interaction with the pointer occurs for a short time, outside of which coupling constant g is zero,
R so that the evolutionary operator is
Û = exp(− ~i Ĥint dt) = exp(− ~i g ÔP̂ ). After the interaction with the pointer, the system undergoes a projective measurement where only a subset of the measured
states are chosen. Labelling this post-selected state |ψf i,
the final pointer state is (~ = 1)
|mf i = hψf | exp(−ig ÔP̂ )|ψi i|mi i
hf |Ô|ii
P̂ |mi i
≈ hψf |ψi i 1 − ig
hψf |ψi i
(2)
≈ hψf |ψi i exp(−ighÔiw P̂ )|mi i
where
hÔiw ≡
hψf |Ô|ψi i
hψf |ψi i
is known as the weak value of Ô.
(3)
2
The pointer momentum P̂ is conjugate to the pointer
position Q̂. Let us now write
R the initial pointer state
in the position basis, |mi i = dq|qiϕ(q), where ϕ(q) ≡
hq|mi i. The final pointer state in the position basis then
is
Z
|mf i ≈ hψf |ψi i exp(−ighÔiw P̂ ) dq|qiϕ(q)
Z
(4)
= hψf |ψi i dq|qiϕ(q − ghÔiw )
where we have used the fact that P̂ acts as a translation
operator that shifts the pointer state in the conjugate
q-basis by ghÔiw .
What this means is that if the pointer states were the
positions of a needle on a measuring device, the interaction of the measurement device with the system will shift
the position of the needle by a distance proportional to
hÔiw , thereby giving us a measurement of observable Ô.
Double-slit experiment. The interference pattern in the
double-slit experiment is due to the phase differences φ
of the paths emanating from the slits. Let |Ai and |Bi
form an orthonormal basis representing the state of particles that went through slit A and B. For a point on
the screen identified by √
the angle θ in Fig. 1, the state
is |ψi = (|Ai + eiφ |Bi)/ 2 (we work in the regime were
higher-order paths which pass through both slits are negligible [19]). φ = kd sin θ, where k is the wavenumber
of the light source and d is the slit spacing. We will
as usual assume that the double-slit experiment is in a
steady-state.
The polarisation of the photon is described by a linear combination of the horizontal |Hi and vertical |V i
basis states. We also√define right-circular polarisation
|Ri ≡ (i|Hi + |V i)/ 2, √and left-circular polarisation
states |Li ≡ (i|Hi − |V i)/ 2.
For coherent interference, the particles from slit A and
B must have the same polarisation. Without loss of generality we assume right-circular polarisation,
√
|ψo i = (|Ai + eiφ |Bi)|Ri/ 2 .
(5)
Projecting onto a position basis (for clarity, we distinguished the pointer position basis |qi from the system
position basis |xi), the probability of detection of the
particle at the detection screen is
P (x) = hx|ψo ihψo |xi
1
= (|ψA (x)|2 + |ψB (x)|2 + 2|ψA (x)||ψB (x)| cos φ) .
2
(6)
This produces the well known interference pattern of
Young’s double-slit experiment. The cos φ term encapsulates the interference component of the pattern: destructive interference occurs when the √
paths are out of phase
[φ = (2n+1)π], |−i ≡ (|Ai−|Bi)/ 2; constructive interference occurs wihen
√ the paths are in phase (φ = 2nπ),
|+i ≡ (|Ai + |Bi)/ 2, where n is an integer.
One can non-destructively reveal these phase differences by performing a weak measurement between the
slits and the detection screen. We measure the interference with the following detector for vertical polarisation:
σ̂V = (|+ih+| − |−ih−|)|V ihV |
(7)
The detector for horizontal polarisation σ̂H , is similarly defined. The non-trivial eigenvalues of the detector
are -1 for destructive interference and 1 for constructive
interference. The effect of σ̂V on the state is to switch the
|Ai and |Bi states of the vertical-polarisation component
(similarly for σ̂H ),
σ̂V (cA |Ai + cB |Bi)|V i = (cA |Bi + cB |Ai)|V i,
(8)
where ci are complex co-efficients.
The expectation value of σ̂V reveals the interference
component of the double slit experiment:
hψo |σ̂V |ψo i = cos φ .
(9)
Operationally, if σ̂V was a localised measuring device,
as it moved up and down a plane parallel to the slit plane,
it would measure interference as shifts in its pointer between and including 1 and -1, depending on its position
as determined by angle φ. Note that σ̂H will measure the
same values, hψo |σ̂H |ψo i = cos φ.
Phantom slit. Now we close slit B so that the initial
state is (|Ai|Ri ≡ |A, Ri)
|ψi i = |A, Ri .
(10)
Let us choose the post-selection state
√
|ψf i = (|A, Hi − ieiφ |B, V i)/ 2 .
(11)
From Eq. (3), the weak value of σ̂V and σ̂H for this
system is
hσ̂V iw = cos φ ,
hσ̂H iw = 0 .
(12)
We have taken only the real component of the weak values, as this is the value that is represented by the shift in
the pointer; the imaginary component represents the momentum of the pointer [1]. Another (more operational)
interpretation of the imaginary component is given in
Ref. [20]. Eq. (12) shows that even though slit B does not
physically exists, under the weak measurement scheme,
we can detect an interference pattern as if it does exists and is vertically polarised. In other words, if σ̂V
were a localised measure device, as it as it moved up
and down a plane parallel to the slit plane, it would successively measure constructive and destructive interference, in exactly the same pattern as in the double-slit
experiment [Eq. (9)]. The key difference here however,
is that only measurements which correspond to the postselection state |ψf i are selected. σ̂H in contrast would
register no interference, and therefore the interference
3
polarisation of the photons (V̂ ≡ |V ihV |, Ĥ ≡ |HihH|):
hV̂ iw = 0 ,
(a)
(b)
BS
PBS
BS
BS
PS
HWP
A
PS
B
P2
BS
P1
|ψi〉
(c)
σV
V
H
σH
φ=0
π/2
π
3π/2
2π
FIG. 2. (a) An interferometer setup of the double-slit experiment. A beam splitter (BS) splits the light into two paths,
representing the paths from two slits. A phase shifter (PS)
introduces a phase difference eiφ between the two arms. A
final BS combines the two paths to coherently interfere, as
detected by the (CCD) camera. (b) An interferometer setup
of the phantom slit effect. Polarisers at P1 implement σ̂V
and σ̂H ; polarisers at P2 implement V̂ and Ĥ. A half-way
plate (HWP), PS, BS, and polarising beam-splitter (PBS)
ensure that states orthogonal to |ψf i are not selected. The
measurements are weak, so that the original source beam is
undisturbed (dotted line) by the polarisers. (c) depicts the
spot size at a given level of intensity as detected by a camera
in (b), if all four weak measurements (σ̂V , σ̂H , V̂ , Ĥ) were applied simultaneously for different values of φ. The elements
constituting σ̂V (σ̂H ) slightly deflects the beam up (down),
and V̂ (Ĥ) slightly deflects the beam left (right). hσ̂V iw is
at a maximum at φ = 0, 2π (constructive interference) and
disappears at φ = π (destructive interference). Whilst the
interference pattern is vertically polarised, the photons are
horizontally polarised, hĤiw = 1. Note hσ̂H iw = hV̂ iw = 0
pattern comes from the vertical polarisation component
only.
In phantom slit effect, although the intermediate states
exhibit interference, the initial and final states remarkably contain no interference terms: Pi (x) = |ψA (xi )|2 ,
Pf (xf ) = 21 (|ψA (xf )|2 + |ψB (xf )|2 ). Fig. 1 schematically shows the corpuscular nature of the initial and final
states, whilst the intermediate states are wave-like.
Another important feature arises when we measure the
hĤiw = 1 .
(13)
Eq. (13) shows that the photons of the intermediate
states are horizontally polarised. Weak measurements
are a non-destructive process, and therefore all measurements commute and can be made simultaneously. Therefore, whilst the interference pattern of the intermediate
state is vertically polarised, the photons themselves are
simultaneously measured to be horizontally polarised.
Interferometer. The double-slit experiment may be implemented as an interferometer. A beam splitter (BS)
splits the light into two paths, representing the paths
from two slits. The phase differential can be controlled
with a phase shifter (PS) which introduces a phase difference eiφ between the two arms [Fig. 2(a)]. Viewing the
phantom slit effect in an interferometer helps remove the
mysteries of the effect, and weak values in general.
To implement the phantom slit effect one needs to be
able to post-select the state |ψf i. |ψf i is in fact the
QCC post-selection state generalised with a phase differential. One may consider the phantom slit effect as
the dual of the QCC effect, with the roles of polarisation and position reversed. The corresponding schematic
of an interferometric setup for post-selection is given in
Fig. 2(b). The post-selection state is selected with a halfway plate (HWP), PS, BS, and polarising beam splitter
(PBS). The HWP switches polarisation |Hi ↔ |V i. The
PS shifts the phase by ie−iφ . The PBS transmits horizontal polarisation and reflects vertical polarisation. With
path A (B) in Fig. 2(b) corresponding to state |Ai (|Bi),
one sees that under these transformations, the cameras
in Fig. 2(b) detects |ψf i with 100% probability.
In an interferometric setup, σ̂V (σ̂H ) can be implemented as a polariser that reflects vertical (horizontal)
polarisation, placed at location P1 in Fig. 2(b). The vertical (horizontal) polariser is slightly tilted up (down)
to produce a small vertical displacement of the beam.
Similarly, a polariser that reflects vertical (horizontal)
polarisation would implement V̂ (Ĥ), placed at location
P2 in Fig. 2(b). However here the vertical (horizontal)
polariser is slightly tilted left (right) to produce a small
horizontal displacement of the beam. Placed at these
locations, these polarises have the desired effect on the
initial state, i.e.
√
√
σ̂H |ψi i = |BHi/ 2 ,
(14)
σ̂V |ψi i = |BV i/ 2 ,
√
√
Ĥ|ψi i = |AHi/ 2 ,
V̂ |ψi i = |AV i/ 2 ,
(15)
For weak measurements, the polarisers should only interact weakly, letting most of the light through [dotted
line in Fig 2(b)]. Furthermore, the deflection brought
about by these polarisers should be less than the characteristic cross-section width of the beam, so that it is uncertain whether an individual photon has been deflected
4
of not. The detection screen could be charge-coupled device cameras, as for example used in the Koscsis et. al
double-slit interferometer experiment; the relative intensity at the detection screen acts as the pointer.
The tilt of the polarisers are chosen so that the deflected beam is incident on a location of the source beam
spot that is of approximately the same order of intensity.
Therefore after recombination with the source beam, the
normalised intensity detected by the cameras above the
background intensity is given by Eq. (12) and Eq. (13) for
the corresponding detectors. Fig. 2(c) depicts the spot
sizes at a given level of intensity above the background, if
all four weak measurements were applied simultaneously
for different values of φ.
The interpretation of weak values remains an intensely
debated topic. The phantom slit effect should open up
further discussion in the context of the double-slit experiment, which serves as a golden-standard of foundational
quantum interpretation. In the Kocsis et al. double-slit
experiment, weak measurements were used to map the
individual paths of photons, as they moved from the slits
to the detection screen. In further theoretical and experimental work, it would be interesting to map the weak
value trajectories of individual photons of the phantom
slit effect, along the lines of the Kocsis et al. experiment.
Acknowledgements. The authors thank M. Bera, A.
Riera, and S. Quach for discussions and checking the
manuscript. This work was financially supported by
the European Commission’s Marie Curie Actions for Cofunding of Regional, National and International Programmes (COFUND), the Spanish Ministry of Economy and Competitiveness (MINECO), Fundacio Privada
Cellex, Spanish Ministerio de Economı́a y Competitividad grants (Severo Ochoa SEV-2015-0522 and FOQUS
FIS2013-46768-P), and Generalitat de Catalunya (2014
SGR 874).
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∗
[1]
[2]
[3]
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quach.james@gmail.com
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