Rencontres de Moriond, 29 March 2019

Quantum mechanics

and the equivalence principle

Albert Roura

based on arXiv:1810.06744

Part I

Gravitational redshiftin quantum-clock interferometry

Part II

Tests of universality of free fall with non-trivial quantum states

PART I

Gravitational redshiftin

quantum-clock interferometry

arXiv:1810.06744

Relativistic effects in macroscopically delocalized

quantum superpositions

• Macroscopically delocalized quantum superpositions: coherent superposition of atomic wave packets

• Differences in dynamics of superposition components entirely Newtonian

• Same relativistic effects on superposition components (e.g. atomic clocks)

★ Goal (QM + GR): experiment with general relativistic effects acting non-trivially on the quantum superposition

Wavepacket separation

90 ħk beam splitters, sequential two-photon Bragg transitions

Interferometer duration 2T = 2.08 s

TK, P. Asenbaum, C. Overstreet, C. Donnelly, S. Dickerson, A. Sugarbaker, J. Hogan, and M. Kasevich, Nature 2015

Kovachy et al., Nature (2015)

Proper time as which-way information

• Quantum superposition of clocks (COM + internal state) experiencing different proper times

• reduced visibility of interference signal

Zych et al., Nat. Comm. (2011)

| ai+ | biI

II

t

z

a

b

Proper time as which-way information

• Quantum superposition of clocks (COM + internal state) experiencing different proper times

• reduced visibility of interference signal

Zych et al., Nat. Comm. (2011)

I

II

t

z

a

b

���h�(⌧b)|�(⌧a)i��� < 1

| ai ⌦ |�(⌧a)i+ | bi ⌦ |�(⌧b)i

Outline

1. Relativistic effects in macroscopically delocalized quantum superpositions

2. Key elements of quantum-clock interferometry

3. Major challenges in quantum-clock interferometry

4. Doubly differential scheme for gravitational-redshift measurements

5. Feasibility and extensions

Key elements of quantum-clock interferometry

Quantum-clock model

• Quantum overlap:

• Initialization pulse:

• Evolution:

|gi !���(0)

↵=

1p2

⇣|gi+ i ei'|ei

⌘

|gi

|ei

�E

���h�(⌧b)|�(⌧a)i��� = cos

✓�E

2~ (⌧b � ⌧a)

◆

���(⌧)↵/ 1p

2

⇣|gi+ i ei'e�i�E ⌧/~|ei

⌘

• Comparison of independent clocks (after read-out pulse):

• Instead of independent clocks we pursue a quantum superposition at different heights.

a

b

t

z

Lz ⇠ 1 cm�E ⇠ 1 eV�⌧b ��⌧a ⇡

�g Lz/c

2��t

for optical atomic clocks

a

b

t

z

Lz

• Comparison of independent clocks (after read-out pulse):

• Instead of independent clocks we pursue a quantum superposition at different heights.

a

b

t

z

Lz ⇠ 1 cm�E ⇠ 1 eV�⌧b ��⌧a ⇡

�g Lz/c

2��t

for optical atomic clocks

a

b

t

z

Lz

• Theoretical description of the clock

‣ two-level atom (internal state):

‣ classical action for COM motion:

H = H1 ⌦ |gihg| + H2 ⌦ |eihe|

Sn

⇥x

µ(�)⇤= �mnc

2

Zd⌧ = �mnc

Zd�

r�gµ⌫

dx

µ

d�

dx

⌫

d�

(n = 1, 2 )

m2 = mg +�m

m1 = mg

�m = �E/c2

Sn

⇥x

µ(�)⇤= �mnc

2

Zd⌧ �

Zd⌧ Vn(x

µ)

free fall

includingexternal forces

• Wave-packet evolution in terms of

‣ central trajectory (satisfies classical e.o.m.)

‣ centered wave packet

• Fermi-Walker frame associated with the central trajectory

‣ valid for freely falling wave packet (geodesic)

‣ but also with external forces /guiding potential (accel. trajectory)

‣ approximately non-relativistic dynamics for centered wave packet

Atom interferometry in curved spacetime (including relativistic effects)

�p/m ⌧ c �x ⌧ `

curvature radius

Xµ(�)

�� (n)c (⌧c)

↵

• Metric in Fermi-Walker coordinates:

• Expanding the action for the centered wave packet:

ds

2 = gµ⌫dxµdx

⌫ = g00 c2d⌧

2c + 2 g0i c d⌧c dx

i + gij dxidx

j

Sn

⇥x(t)

⇤⇡

Zd⌧c

�mnc

2 � Vn(⌧c,0) +mn

2v

2 � 1

2x

T⇣V(n)(⌧c)�mn�(⌧c)

⌘x � V (n)

anh.(⌧c,x)

�

Xµ(⌧c) =�c ⌧c ,0

�

g00 = ��1 + �ij a

i(⌧c)xj/c2

�2 �R0i0j(⌧c,0)xixj +O

�|x|3

�

g0i = �2

3R0jik(⌧c,0)x

jxk + O�|x|3

�

gij = �ij �1

3Rikjl(⌧c,0)x

kxl + O�|x|3

�

• Hamiltonian:

• Wave-packet evolution:

‣ propagation phase

‣ centered wave packet

i~ d

d⌧c

�� (n)c (⌧c)

↵= Hc

�� (n)c (⌧c)

↵

Sn = �Z ⌧2

⌧1

d⌧c�mnc

2 + Vn(⌧c,0)�

Hn = mnc2 + Vn(⌧c,0) + H(n)

c

H(n)c =

1

2mnp

2 +1

2x

T⇣V(n)(⌧c)�mn�(⌧c)

⌘x

�� (n)(⌧c)↵= eiSn/~ �� (n)

c (⌧c)↵

V(n)ij (⌧c) = @i@jVn(⌧c,x)

��x=0

• Full interferometer (including laser kicks):

• Detection probability at the exit port(s):

• Phase shift:

h I| Ii =1

2

�1 + cos ��

�

�� = �b � �a + ��sep

I

II

a

b

t

z

propagation + laser phases

| Ii =1

2

�ei�a + ei�b

�| ci

Major challenges in quantum-clock interferometry

Insensitivity to gravitational redshift (in a uniform field)

• Consider a freely falling frame:

• Proper-time difference between the two interferometer branches independent of

• (small dependence due to pulse timing suppressed by )

g

(vrec/c) ⇠ 10�10

Ramsey-Bordéinterferometer

I

II

a

b

t

z

⌧c

Insensitivity to gravitational redshift (in a uniform field)

• Consider a freely falling frame:

• Proper-time difference between the two interferometer branches independent of

• (small dependence due to pulse timing suppressed by )

g

(vrec/c) ⇠ 10�10

Ramsey-Bordéinterferometer

I

II

a

b

t

z

⌧c

Differential recoil

• Different recoil velocities different central trajectories

• Implied changes of proper-time difference are comparableto signal of interest.

I

II

a

b

t

z

⌧c

v(n)rec = ~ke↵ /mn

Small visibility changes

• Reduced interference visibility due to deceasing quantum overlap of clock states:

• Small effect for feasible parameter range:

• Extremely difficult to measure such small changes of visibility, which are masked by other effects leading also to loss of visibility.

���h�(⌧b)|�(⌧a)i��� = cos

✓�E

2~ (⌧b � ⌧a)

◆

���h�(⌧b)|�(⌧a)i��� = cos

✓!0

2

g�z

c2�t

◆⇡ 1�

�10

�3�2/2

�E/~ = !0 ⇡ 2⇡ ⇥ 4⇥ 102 THz

�z = 1 cm

�t = 1 s

h I| Ii =

1

2

+

1

2

���h�(⌧b)|�(⌧a)i��� cos ��

Small visibility changes

• Reduced interference visibility due to deceasing quantum overlap of clock states:

• Small effect for feasible parameter range:

• Extremely difficult to measure such small changes of visibility, which are masked by other effects leading also to loss of visibility.

���h�(⌧b)|�(⌧a)i��� = cos

✓�E

2~ (⌧b � ⌧a)

◆

�E/~ = !0 ⇡ 2⇡ ⇥ 4⇥ 102 THz

�t = 1 s

�z = 1m���h�(⌧b)|�(⌧a)i

��� = cos

✓!0

2

g�z

c2�t

◆⇡ 1�

�10

�1�2/2

h I| Ii =

1

2

+

1

2

���h�(⌧b)|�(⌧a)i��� cos ��

Doubly differential scheme for gravitational-redshift measurement

• Detection probability at first exit port (independent of internal state):

• Phase-shift difference directly related to visibility reduction.

• Precise differential phase-shift measurement involving state-selective detection is much more viable.

• ( immune to spurious loss of contrast + common-mode rejection of phase noise)

Differential phase-shift measurement

visibility

h I| Ii =1

2

⇣⌦

(1)I

��

(1)I

↵+

⌦

(2)I

��

(2)I

↵⌘

=

1

4

⇣1 + cos ��(1)

+ 1 + cos ��(2)⌘

=

1

2

+

1

2

cos

✓��(2) � ��(1)

2

◆cos

✓��(1)

+ ��(2)

2

◆

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• Level structure for group-II-type atoms (e.g. Sr, Yb) employed in optical atomic clocks:

• Two-photon process resonantly connecting the two clock states.

• Equal-frequency counter-propagating laser beams in lab frame:

• constant effective phase simultaneity hypersurfaces in lab frame

Two-photon pulse for clock initialization

E. A. ALDEN, K. R. MOORE, AND A. E. LEANHARDT PHYSICAL REVIEW A 90, 012523 (2014)

1S0

1P1

3P13P0

3P2

E1

M1 ∆

3S1

E1

FIG. 1. (Color online) Two-photon clock level structure. This isthe prototypical, optical clock-level structure with a 3P0 clock state.The electric field of one photon and the magnetic field of a seconddegenerate photon directly couple the 1S0 ground state to the clockstate by coupling to the intermediate 3P1 level with some detuning !.

A sample detection channel for a hot clock is the 3P0E1←→ 3S1, E1

allowed transition.

require optimization are vapor cell temperature T and laserbeam radius ω0. The effective rate of atoms excited to the 3P0level is given by

N3P0 = P3P0 (T ,ω0)Ntot(T ,ω0), (3)

where P3P0 is the probability a single atom in the excitationregion has been excited to the 3P0 level and Ntot is the rate ofatoms flowing through the interrogation region. In a thermalenvironment the interrogation time of an atom by the excitationlaser is always much less than the time required to coherentlytransfer the full population to the excited state; there is norisk of Rabi flopping. An increase in laser power is thereforealways beneficial because it increases the two-photon Rabifrequency and by extension the probability of excitation to the3P0 level in a time-limited measurement. The temperature andlaser beam radius contribution to overall rates and stabilitywill be explained in Sec. III B.

A. Two-photon Rabi frequency

For the purposes of this paper, we will consider a systemwhere the atom is excited with a single laser (monochromatic)or pair of lasers (bichromatic) whose frequencies are far

LoT

2ω0

2zR

FIG. 2. (Color online) Hot optical clock. This diagram of amonochromatic laser in a vapor cell depicts the experimental system.The detection length Lo is the detection optics aperture, and the laserbeam radius is ω0. The Rayleigh range zR that limits the interrogationregion is shown in this graphic. The area enclosed by a box is theRayleigh-limited interrogation region of the atoms.

off resonance from the allowed E1 or M1 transitions. Tosatisfy the selection rules of the transition, the electric fieldvector of one excitation photon must be parallel to themagnetic field vector of the other excitation photon. Thisalignment can be realized utilizing either the Lin ⊥ Linor σ+/σ− polarization scheme described by Dalibard andCohen-Tannoudji [9]. These schemes satisfy the selection rulesand ensure that any clock excitation is the product of excitationfrom counterpropagating beams and thus reduces or eliminatesfirst-order Doppler broadening.

Our proposed system satisfies the constraints of adiabaticelimination [10,11], specifically ! ≫ $1,$2,δ, where $i isthe two-level Rabi frequency of each E1 and M1 transition,δ is the two-photon detuning from the unperturbed transitionfrequency, and ! is the minimum detuning of an excitationphoton’s energy from the intermediate 3P1 level (see Fig. 1).In this limit, the two-photon Rabi frequency for an atomaddressed by a pair of photons, where δ is chosen to offsetthe light shift, is given by [12]

$R2γ = 2I

!2c2ϵ0

⟨3P0||µ||3P1⟩M1⟨3P1||D||1S0⟩E1

!, (4)

where I is the peak intensity of the excitation laser,⟨3P0||µ||3P1⟩M1 is the reduced matrix element for the magneticdipole (M1) transition, and ⟨3P1||D||1S0⟩E1 is the reducedmatrix element for the electric dipole (E1) transition.

The E1-M1 coupling will also occur via the 1P1 inter-mediate level. In the case of Hg it will constitute as muchas 37% of the Rabi frequency, where its contribution ismaximum for the degenerate excitation scheme. We omit thisfavorable contribution from the rate and stability simulationsfor simplicity, but experiments can anticipate an enhancement.Estimated and observed electric and magnetic dipole matrixelements are shown for the group-II-type atoms in Table I.We also provide the estimated two-photon Rabi frequency forthe degenerate photon case with the experimental parametersdefined in Table II.

TABLE I. Reduced matrix elements for the electric dipole⟨nsnp3P 1||D||ns2 1S0⟩ intercombination transition (E1) and the mag-netic dipole ⟨nsnp3P 0||µ||nsnp3P1⟩ transition (M1) for each candi-date element. Matrix element values are in a.u. For monochromaticexcitation, the two-photon Rabi frequency $R2γ is shown for unitintensity (1 W/m2). A prototypical intensity for this scheme is6 × 106 W/m2.

Atom n E1/ea0 M1/µB

$R2γ /I (Hz)

Ra 7 1.2 [13]√

2 [14] 7.1 × 10−5

Ba 6 0.45 [15]√

2 [14] 3 × 10−5

Yb 6 0.54 [16]√

2 [16] 2.5 × 10−5

Hg 6 0.44 [17]√

2 [14] 9.3 × 10−6

Sr 5 0.15 [18]√

2 [14] 8.8 × 10−6

Ca 4 0.036 [18]√

2 [14] 2 × 10−6

Mg 3 0.0057 [18]√

2 [17] 2.2 × 10−7

Be 2 0.00024 [17]√

2 [14] 9.3 × 10−9

012523-2

Ald

en e

t al.,

Phys

. Rev

. A (2

014)

|gi

|ei

ei!teik·x ⇥ ei!te�ik·x = ei 2!t

k<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

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• Compare differential phase-shift measurements for different initialization times:

Laboratory frame

���(2)(t0i)� ��(1)(t0i)

��

���(2)(ti)� ��(1)(ti)

�=

�E

2~ (�⌧b ��⌧a) = �mg�z (t0i � ti)/~

I

II

a

b

t

z

• Compare differential phase-shift measurements for different initialization times:

Laboratory frame

���(2)(t0i)� ��(1)(t0i)

��

���(2)(ti)� ��(1)(ti)

�=

�E

2~ (�⌧b ��⌧a) = �mg�z (t0i � ti)/~

ti

I

II

a

b

t

z

• Compare differential phase-shift measurements for different initialization times:

Laboratory frame

I

II

a

b

t

z t0iti

���(2)(t0i)� ��(1)(t0i)

��

���(2)(ti)� ��(1)(ti)

�=

�E

2~ (�⌧b ��⌧a) = �mg�z (t0i � ti)/~

• Relativity of simultaneity:

Freely falling frame

���(2)(t0i)� ��(1)(t0i)

��

���(2)(ti)� ��(1)(ti)

�=

�E

2~ (�⌧b ��⌧a) = �mg�z (t0i � ti)/~

I

II

a

b

t

z

⌧c

�⌧c ⇡ �v(t)�z/c2 = g (t� tap)�z/c2

Challenges addressed

• Differential phase-shift measurement precise measurement, common-mode rejection (of noise & systematics)

• Comparing measurements with different initialization times sensitive to gravitational redshift + further immunity

• Almost no recoil from initialization pulse, small residual recoil with no impact on gravitational redshift measurement,

• effect of differential recoil from second pair of Bragg pulses cancels out in doubly differential measurement.

• Residual recoil with no influence on the phase-shift for the excited state:

I

II

a

t

z

b

Feasibility and extensions

• 10-m atomic fountains operating with Sr, Yb in Stanford & Hannover respectively.

• More than of free evolution time.

• Doubly differential phase shift of for

• Resolvable in a single shot for atomic clouds with atoms (shot-noise limited)

• More compact set-ups possible with guided or hybrid interferometry (less mature).

Feasible implementation

HITec (Hannover)

2 s

1mrad

N = 106

�E/~ = !0 ⇡ 2⇡ ⇥ 4⇥ 102 THz

�z = 1 cm

�ti = 1 s

Conclusion

• Measurement of relativistic effects in macroscopically delocalized quantum superpositions with quantum-clock interferometry.

• Important challenges in quantum-clock interferometry and its application to gravitational-redshift measurement.

• Promising doubly differential scheme that overcomes them.

• Feasible implementation in facilities soon to become operational.

• Applicable also to more compact set-ups based on guided or hybrid interferometry.

• If one considers a consistent framework for parameterizing violations of Einstein’s equivalence principle, (e.g. dilaton models)

• both for comparison of independent clocks and for the above quantum-clock interferometry scheme one obtains

• test of universality of gravitational redshift with delocalized quantum superpositions

↵e-g =m1

�m

��2 � �1

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�⌧b ��⌧a�⌧a

⇡�1 + ↵e-g

�⇣U(xb)� U(xa)

⌘/c2

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PART II

Tests of universality of free fall with

non-trivial quantum states

• Test of UFF with differential atom interferometry.

• Comparing the gravitational acceleration of the two hyperfine levels of the electronic ground state of :

• Comparing also with the acceleration for an atom in a coherent superposition of those internal states:

Quantum superposition of internal states

G. Rosi, G. D’Amico, L. Cacciapuoti, F. Sorrentino, M. Prevedelli, M. Zych, C. Brukner and G. Tino, Nat. Comm. 8 15529 (2017)

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|F = 1, mF = 0i<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

|F = 2, mF = 0i<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

vs.

�|F = 1, mF = 0i + |F = 2, mF = 0i

�/p2

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Two entangled atomic species

ongoing experiments, namely two entangled atoms ofdifferent species. The experiment considers the comparisonof the free-fall acceleration of an atom A when it isentangled with a different atomic species B to the free-fall acceleration of the atoms without entanglement. Wedescribe a particular implementation with 85Rb and 87Rbatoms and an entangling process based on a vacuumstimulated rapid adiabatic passage protocol implementedin a high-finesse optical cavity.The concept of our proposal relies on a vertical atom

interferometer in which atomic species A and B areentangled. The entanglement is heralded at the first beamsplitter of the interferometer by the detection of a singlephoton. The scheme is related to the seminal work inRefs. [26,27], but operates here on freely propagating,distinguishable atoms instead of trapped, identical particles.In the event of the emission of a single photon from one ofthe two atoms in the direction of a photon detector, andassuming that it is not possible to distinguish which atomemitted the photon, a detection event will herald a super-position state: Atom A acquires the momentum ℏk (Aemitted the photon of wave vector k) and atom B is leftunperturbed, or vice versa. The corresponding entangledstate can be written as

jψi ¼ 1ffiffiffi2

p ðjA;ℏk;B; 0iþ eiϕjA; 0;B;ℏkiÞ: ð1Þ

The beam splitter thus creates a superposition of themomenta of the two atomic species A and B, with ϕ afixed (nonrandom) phase in the case of a coherent super-position. To complete the interferometer, the two pathsproduced at the first beam splitter are subsequently manip-ulated with conventional atom optics (e.g., two-photonRaman transitions [28]) in order for the paths of eachspecies to interfere. Single atom detectors are finally usedto probe the atomic interference at the interferometer output.We focus in this Letter on a particular implementation of

this idea using 85Rb and 87Rb atoms, as sketched in Fig. 1.To entangle the two atoms, we propose to employ a vacuumstimulated Raman adiabatic passage (vSTIRAP) protocol[29], where the detection of a single photon exiting a high-finesse optical ring cavity heralds the entangled state ofEq. (1). The cavity is on resonance with a mode offrequency ωc. The two atoms are initialized in one of theirtwo hyperfine ground states, respectively, jF ¼ 3i for A ¼85Rb and jF ¼ 2i for B ¼ 87Rb; see Fig. 1(b). ThevSTIRAP process is triggered at time t ¼ t0 by a pulseof two pump laser beams at frequenciesωA

p andωBp (red and

blue vertical arrows), which fulfill the two-photon Ramanresonance condition for each atom: ωα

p − ωc ¼ Gα þ ωαR,

where Gα is the hyperfine splitting frequency, and ωαR is the

two-photon recoil frequency, with α ¼ A, B ¼ 85Rb, 87Rb[30]. Assuming that the probability of the adiabatic passagefor each atom is small [26,27], the vSTIRAP process willin all likelihood deposit at most a single photon into thecavity. The photon can then escape the cavity while one of

the atoms is transferred from one hyperfine state to theother [29,31]. If the photon emission of both atomic speciescan be made to have the same envelope and frequency, thena detection event will herald the desired entangled state.In view of the WEP test, we aim to measure the

gravitational acceleration with the atom interferometer,requiring a vertical accelerometer [28]. Therefore, at leastone of the light beams realizing the Raman transition musthave a projection on the gravity direction (z). We choose aconfiguration where the cavity is horizontal (xy planein Fig. 1) and where the pump beams are aligned withgravity. As a consequence, the beam splitter operates intwo dimensions, with a transfer of momentum ℏktot ≡ℏðkxx − kzzÞ along the x and z direction, with kx ¼ ωc=c(kz) the wave vector of the cavity (pump) photon. Theremaining part of the interferometer is a typical Mach-Zehnder configuration [28], apart from the fact that themirror and final beam splitter pulses are two-dimensional inthe momentum transfer; see Fig. 1(c).

(a)

(c)

(b)

FIG. 1. Implementation with 85Rb and 87Rb atoms and avSTIRAP protocol to realize the entangling beam splitter. (a) Gen-eral sketch of the experiment: the atoms are laser cooled andthen released in a high-finesse optical cavity made of three mirrorslying in the ðxyÞ plane. During the vSTIRAP process, a photon isextracted from the pump beam (red and blue arrows for 85Rb and87Rb, respectively), and a photon is emitted into the cavity mode.The emitted photon (frequency ωc) is detected at one output of thecavity (“click”). (b) Energy levels of the atoms subject to two-photon Raman transitions. The high-finesse cavity is resonant for amode of frequency ωc. The vSTIRAP process is initiated at timet ¼ t0 by a pulse of the pump beams of frequency ωA;B

p . The grayarrow represents a laser beam (frequencyωx ¼ ωc) used to performthe Raman transitions in the mirror pulse and final beam splitterpulse of the interferometer. (c) Space-time diagrams of the atominterferometer in the x and z directions. In (a) and (c), the differencein recoil velocities between 85Rb and 87Rb has been exaggerated to10% (instead of 2.3%). In the bottom of (c), gravity has beenreduced to g ¼ 0.01 ms−2 in order to highlight the recoil effect.

PHYSICAL REVIEW LETTERS 120, 043602 (2018)

043602-2

R. Geiger and M. Trupke, Phys. Rev. Lett. 120 043602 (2018)

• Collaboration on the experimental realization of the proposed scheme with Leibniz University Hannover :

• Related theoretical work at Ulm University :

Sina Loriani Dennis Schlippert Ernst Rasel

Acknowledgments

Stephan Kleinert Christian Ufrecht

PHYSICAL REVIEW A 99, 013627 (2019)

Proper time in atom interferometers: Diffractive versus specular mirrors

Enno Giese,1,* Alexander Friedrich,1 Fabio Di Pumpo,1 Albert Roura,1 Wolfgang P. Schleich,1,2

Daniel M. Greenberger,3 and Ernst M. Rasel41Institut für Quantenphysik and Center for Integrated Quantum Science and Technology (IQST), Universität Ulm,

Albert-Einstein-Allee 11, D-89069 Ulm, Germany2Hagler Institute for Advanced Study and Department of Physics and Astronomy, Institute for Quantum Science and Engineering (IQSE),

Texas A&M AgriLife Research, Texas A&M University, College Station, Texas 77843-4242, USA3City College of the City University of New York, New York, New York 10031, USA

4Institut für Quantenoptik, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany

(Received 19 September 2018; revised manuscript received 12 December 2018; published 30 January 2019)

We compare a conventional Mach–Zehnder light-pulse atom interferometer based on diffractive mirrors withone that uses specular reflection. In contrast to diffractive mirrors that generate a symmetric configuration,specular mirrors realized, for example, by evanescent fields lead under the influence of gravity to an asymmetricgeometry. In such an arrangement the interferometer phase contains nonrelativistic signatures of proper time.

DOI: 10.1103/PhysRevA.99.013627

I. INTRODUCTION

The redshift controversy [1] has triggered a lively debate[2–7] about the role of proper time in atom interferometers.Unfortunately, the discussion was focused solely on a light-pulse Mach–Zehnder interferometer (MZI) where, due to thesymmetry of the interferometer, the proper-time differencevanishes [8]. However, this symmetry depends crucially onthe way the mirrors change the atomic trajectory. In thepresent article we propose an interferometer geometry calledthe specular mirror interferometer (SMI) where the proper-time difference does not vanish [8–10] due to the specificnature of the mirror.1

Such configurations are crucial in studying proper-timeeffects in atom interferometers [11] experimentally [12,13]2

and might be used for other tests of the foundations of physicssuch as the equivalence principle [14,15].

A. The role of the mirror

The symmetry of an MZI is intrinsically linked to thediffractive nature of the mirror pulses and has to be broken inorder to observe nonvanishing proper-time contributions to themeasured phase. One possibility is the use of Ramsey–Bordé-type configurations [16,17], where diffractive beam splittersare applied asymmetrically to both branches. Here, we pro-

*enno.a.giese@gmail.com1The crucial role of mirrors in determining the phase was already

pointed out in Ref. [8]. Even though the claim that neutron inter-ferometers [10] are operated with specular mirrors was challengedby Lemmel in Ref. [9], the conclusion that specular mirrors lead toa nonvanishing proper time is valid and underlined in the presentarticle.

2References [12,13] describe experiments where the effect ofproper-time differences between interferometer paths is simulated bydifferent magnetic gradients.

pose an alternative geometry that relies on specular mirrors[18] inverting the incoming momentum. When combined withthe influence of gravity, the specular nature of the mirrorsleads to an asymmetry that causes a proper-time contributionto the interferometer phase.

There exist several proposals to use specular reflection atevanescent fields to build a cavity for atoms and in whichlinear gravity is taken into account [19]. In contrast to theseideas and Fabry–Pérot atom interferometers [20], in whichthe atoms are localized over the length of the interferometer,we use specular mirrors not to confine atoms but to inves-tigate the output ports of a two-branch interferometer. Foran overview of specular mirrors based on evanescent fields,we refer to Refs. [21,22]. As an alternative to evanescentfields, strong magnetic mirrors and even permanent magneticstructures can be used for atom optics [23,24].

B. Overview

Our analysis proceeds in three steps: (i) In Sec. II wecompare the MZI and SMI in a semiclassical description anddiscuss the emergence of the total phase in the laboratoryframe as well as in a frame freely falling with the atoms. (ii)We then resort in Sec. III to a representation-free description[25,26] of both interferometers by introducing an operatorto describe the specular reflection. (iii) Finally, we study inSec. IV the reflection of a particle at an exponential potentialand identify our analytical results with the specular reflectionoperator. We also discuss some of the challenges of such aconfiguration in Sec. V before we conclude in Sec. VI. In theappendix we use the operator formalism introduced in Sec. IIIto calculate the interference pattern of an SMI.

II. SEMICLASSICAL DESCRIPTION

Following de Broglie [27] and the path integral formulationof quantum mechanics [28], a particle accumulates the phase! ≡ −mc2

!dτ/h along its path, where m is the mass of

2469-9926/2019/99(1)/013627(9) 013627-1 ©2019 American Physical Society

PHYSICAL REVIEW A 99, 013627 (2019)

Proper time in atom interferometers: Diffractive versus specular mirrors

Enno Giese,1,* Alexander Friedrich,1 Fabio Di Pumpo,1 Albert Roura,1 Wolfgang P. Schleich,1,2

Daniel M. Greenberger,3 and Ernst M. Rasel41Institut für Quantenphysik and Center for Integrated Quantum Science and Technology (IQST), Universität Ulm,

Albert-Einstein-Allee 11, D-89069 Ulm, Germany2Hagler Institute for Advanced Study and Department of Physics and Astronomy, Institute for Quantum Science and Engineering (IQSE),

Texas A&M AgriLife Research, Texas A&M University, College Station, Texas 77843-4242, USA3City College of the City University of New York, New York, New York 10031, USA

4Institut für Quantenoptik, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany

(Received 19 September 2018; revised manuscript received 12 December 2018; published 30 January 2019)

We compare a conventional Mach–Zehnder light-pulse atom interferometer based on diffractive mirrors withone that uses specular reflection. In contrast to diffractive mirrors that generate a symmetric configuration,specular mirrors realized, for example, by evanescent fields lead under the influence of gravity to an asymmetricgeometry. In such an arrangement the interferometer phase contains nonrelativistic signatures of proper time.

DOI: 10.1103/PhysRevA.99.013627

I. INTRODUCTION

The redshift controversy [1] has triggered a lively debate[2–7] about the role of proper time in atom interferometers.Unfortunately, the discussion was focused solely on a light-pulse Mach–Zehnder interferometer (MZI) where, due to thesymmetry of the interferometer, the proper-time differencevanishes [8]. However, this symmetry depends crucially onthe way the mirrors change the atomic trajectory. In thepresent article we propose an interferometer geometry calledthe specular mirror interferometer (SMI) where the proper-time difference does not vanish [8–10] due to the specificnature of the mirror.1

Such configurations are crucial in studying proper-timeeffects in atom interferometers [11] experimentally [12,13]2

and might be used for other tests of the foundations of physicssuch as the equivalence principle [14,15].

A. The role of the mirror

The symmetry of an MZI is intrinsically linked to thediffractive nature of the mirror pulses and has to be broken inorder to observe nonvanishing proper-time contributions to themeasured phase. One possibility is the use of Ramsey–Bordé-type configurations [16,17], where diffractive beam splittersare applied asymmetrically to both branches. Here, we pro-

*enno.a.giese@gmail.com1The crucial role of mirrors in determining the phase was already

pointed out in Ref. [8]. Even though the claim that neutron inter-ferometers [10] are operated with specular mirrors was challengedby Lemmel in Ref. [9], the conclusion that specular mirrors lead toa nonvanishing proper time is valid and underlined in the presentarticle.

2References [12,13] describe experiments where the effect ofproper-time differences between interferometer paths is simulated bydifferent magnetic gradients.

pose an alternative geometry that relies on specular mirrors[18] inverting the incoming momentum. When combined withthe influence of gravity, the specular nature of the mirrorsleads to an asymmetry that causes a proper-time contributionto the interferometer phase.

There exist several proposals to use specular reflection atevanescent fields to build a cavity for atoms and in whichlinear gravity is taken into account [19]. In contrast to theseideas and Fabry–Pérot atom interferometers [20], in whichthe atoms are localized over the length of the interferometer,we use specular mirrors not to confine atoms but to inves-tigate the output ports of a two-branch interferometer. Foran overview of specular mirrors based on evanescent fields,we refer to Refs. [21,22]. As an alternative to evanescentfields, strong magnetic mirrors and even permanent magneticstructures can be used for atom optics [23,24].

B. Overview

Our analysis proceeds in three steps: (i) In Sec. II wecompare the MZI and SMI in a semiclassical description anddiscuss the emergence of the total phase in the laboratoryframe as well as in a frame freely falling with the atoms. (ii)We then resort in Sec. III to a representation-free description[25,26] of both interferometers by introducing an operatorto describe the specular reflection. (iii) Finally, we study inSec. IV the reflection of a particle at an exponential potentialand identify our analytical results with the specular reflectionoperator. We also discuss some of the challenges of such aconfiguration in Sec. V before we conclude in Sec. VI. In theappendix we use the operator formalism introduced in Sec. IIIto calculate the interference pattern of an SMI.

II. SEMICLASSICAL DESCRIPTION

Following de Broglie [27] and the path integral formulationof quantum mechanics [28], a particle accumulates the phase! ≡ −mc2

!dτ/h along its path, where m is the mass of

2469-9926/2019/99(1)/013627(9) 013627-1 ©2019 American Physical Society

QUANTUS group @ Ulm University

Wolfgang Schleich Albert Roura

Wolfgang Zeller Stephan Kleinert

Jens Jenewein

Christian Ufrecht

Sabrina Hartmann

Matthias Meister Enno Giese

Alexander Friedrich Fabio Di Pumpo Eric Glasbrenner

Thank you for your attention.

Q-SENSE European Union H2020 RISE Project