Observation of the Phononic Lamb Shift with a Synthetic Vacuum

Selected for a Viewpoint in Physics PHYSICAL REVIEW X 6, 041041 (2016)

T. Rentrop,1,* A. Trautmann,1 F. A. Olivares,1 F. Jendrzejewski,1 A. Komnik,2 and M. K. Oberthaler1 1Kirchhoff-Institut für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany 2Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany (Received 8 July 2016; revised manuscript received 10 October 2016; published 28 November 2016) In contrast to classical empty space, the quantum vacuum fundamentally alters the properties of embedded particles. This paradigm shift allows one to explain the discovery of the celebrated Lamb shift in the spectrum of the hydrogen atom. Here, we engineer a synthetic vacuum, building on the unique properties of ultracold atomic gas mixtures, offering the ability to switch between empty space and quantum vacuum. Using high-precision spectroscopy, we observe the phononic Lamb shift, an intriguing many-body effect originally conjectured in the context of solid-state physics. We find good agreement with theoretical predictions based on the Fröhlich model. Our observations establish this experimental platform as a new tool for precision benchmarking of open theoretical challenges, especially in the regime of strong coupling between the particles and the quantum vacuum.

DOI: 10.1103/PhysRevX.6.041041 Subject Areas: Atomic and Molecular Physics, Quantum Physics

I. INTRODUCTION which realizes the phononic vacuum. In analogy to the electron bound to a donor ion, we additionally localize the When an electron moves in the vacuum it drags a particles of the minority species in a tight optical trap. By cloud of virtual photons with it [1]. As a result, its mass removing the BEC this system allows for switching off effectively increases. Since the coupling between the the quantum vacuum and with that to directly compare the electron and the electromagnetic field cannot be switched dressed particle to the bare one, a feature that does not exist off, it is impossible to observe the “bare” electron mass in experimental tests of quantum electrodynamics (QED) directly. An observable effect exists for the bound electron [16,17] and semiconductor systems. leading to a shift of the electronic energy levels in the We observe the phononic Lamb shift directly by high- hydrogen atom, known as the Lamb shift [2–5]. Similar resolution spectroscopy of the two lowest energy levels of effects take place in semiconductors where electrons couple the bound particle employing motional Ramsey spectros- to phononic excitations [6,7], leading to a quasiparticle copy [18], which provides improved resolution compared known as a polaron. In these systems, the increase of the to other previously demonstrated spectroscopic techniques effective mass has been observed [8]. However, a quanti- [15,19–21]. In this method, we measure the relative energy tative measurement of the predicted phononic Lamb shift shift of particles confined in an approximately harmonic [9,10], which is defined for an electron bound to a donor potential by letting them oscillate. For weak confinement ion, is still missing. Progress is hindered by uncontrolled we expect a slow down of oscillations due to the increased disorder effects in such solid-state systems [11]. effective mass, which is computed for free particles; see Here, we realize a model system for such a phononic Fig. 1(a). However, the confinement effects go far beyond quantum vacuum based on a strongly imbalanced mixture this simple effective mass concept and have to be explicitly of ultracold atomic gases. Imbalanced Fermi gases already taken into account. We term the deviation between these allowed for the observation of the Fermi polaron [12–15] and two corrections phononic Lamb shift. Importantly, it direct measurement of its increased effective mass [15] via can even lead to an increase of oscillation frequency as oscillations in a weak optical trap. To observe the phononic observed in our experiment. For our weakly interacting Lamb shift, we immerse the particles of the minority species system the phononic Lamb shift is well described within in a weakly confined Bose-Einstein condensate (BEC), the Fröhlich model [22–25] and can be computed with well-controlled perturbative methods. *[email protected]

Published by the American Physical Society under the terms of II. EXPERIMENTAL SETUP the Creative Commons Attribution 3.0 License. Further distri- In our experiment, a BEC of about 106 sodium atoms and bution of this work must maintain attribution to the author(s) and 3 4 the published article’s title, journal citation, and DOI. few 10 to several 10 lithium atoms are both trapped by the

2160-3308=16=6(4)=041041(10) 041041-1 Published by the American Physical Society T. RENTROP et al. PHYS. REV. X 6, 041041 (2016)

(a) (b)

eff. self- bare mass energy

Lamb shift

FIG. 1. Experimental setup. (a) Particles are trapped in a harmonic potential with trapping frequency ω0. The coupling of the particles to the quantum vacuum causes energy shifts of the particles’ motional states. For a free particle, it leads to a mass increase, which would manifest itself to a smaller energy difference between the harmonic trap levels, as shown in the second column of the inset. For a trapped particle, the effective mass approximation breaks down and the energy shift has to be calculated from the self-energy in the confined geometry. The difference between the complete calculation and the effective mass approximation is termed phononic Lamb shift, and it can be so substantial that the interaction with the vacuum actually leads to an increased energy difference compared to the bare case, as shown in the third column of the inset. (b) Scheme of the experimental setup. The two atomic species are trapped in a single dipole trap (red beams). The lithium atoms, which act as the particles in this setup, experience additionally a strong confinement by a deep small- angle optical lattice (green beams), whose minima can be described by a harmonic potential with trapping frequency ω0. Inset: This setup results in a multitude of independent lithium clouds (blue) embedded in a large sodium Bose-Einstein condensate (yellow), which acts as the quantum vacuum. same two beam optical dipole trap (ODT) at a wavelength of Given the large detuning from the D-line transitions of λ ≈ 589 1064 nm, as sketched in Fig. 1(b) [26].Themeantrapping sodium ( res;Na nm), the depth of the optical lattice 23 ω¯ 2π 150 frequency for Na is ¼ × Hz. The temperature of (VV=h < 0.5 kHz) is much weaker for this species and the sample is about 350 nK. We can choose to work with leads only to weak modulation of the BEC with a chemical 6 7 fermionic Li as well as bosonic Li at mean trapping potential of μV=h ≈ 5.5 kHz. frequencies of ω¯ ¼2π×340Hz for 6Li and ω¯ ¼2π×310Hz for 7Li. The interspecies interaction strength is attractive III. PHONONIC LAMB SHIFT FOR 6 for fermionic Li (aPV ¼ −75a0) [27] and repulsive for FERMIONIC PARTICLES 7 bosonic Li (a ¼þ21a0), with a0 being the Bohr radius. 6 PV First, we discuss the case of fermionic Li particles in We use the subscript P for all quantities related to lithium order to be as close as possible to the conventional Lamb particles, while the variables describing the quantum vacuum created by the sodium condensate carry the subscript V.The shift. Our goal is the measurement of the energy difference ω − intraspecies interaction in Li is negligible, as fermions at low ¼ E1 E0 between the particle energies in the ground temperatures do not scatter and the intraspecies scattering and the first excited states of the harmonic oscillator. We length for the bosons is 7a0 [28]. perform the measurement both in the empty space, where An additional optical lattice close to the D-line tran- the BEC is absent, and with the quantum vacuum in place. λ ≈ 671 For the symmetric superposition of the ground and excited sitions for lithium ( res;Li nm) imposes a very strong confinement in one direction for lithium. It consists of states, ω can be extracted from the phase ϕ ¼ ωt during a two intersecting laser beams, leading to a periodicity of free evolution with duration t. This is the idea behind the 1 65 μ ≈ 78 dlat ¼ . m and a typical depth of VP=h kHz for motional Ramsey sequence we employ here [18]. 6 7 Li (VP=h ≈ 81 kHz for Li), where h is the Planck After initial preparation in the ground state of the constant. This corresponds to an energy gap between the harmonic oscillator, we create a coherent superposition 6 first and second energy band of ω0=2π ¼ 27.3 kHz for Li in the two lowest energy states of the harmonic oscillator by 7 (ω0=2π ¼ 26.2 kHz for Li). Because of the depth of the shaking the optical lattice. The resulting Rabi frequency is potential and the large lattice spacing, tunneling between 1.4 kHz [29], and by coupling the states for 187 μs, an lattice sites can be neglected and the minima can be treated equal superposition of ground and excited state is created as independent harmonic oscillators with trapping fre- (π=2 pulse). Figure 2(b) shows a Rabi cycle, which we quency ω0, resulting in multiple realizations of the experi- use for the calibration of the excited fraction η. In the ment in a single experimental cycle. subsequent time evolution, the excited motional state

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(a) this technique a sufficient optical density can be obtained even for very low lithium atom numbers. To extract the population of the harmonic oscillator states, the absorption (b)1 (c) images are integrated transversally to the lattice direction, as shown in Fig. 2(c). In order to extract the relative 0.5 occupation of the three lowest states, atoms in the areas of the three Brillouin zones are counted and the ratios are 0 00.5 calculated. time[ms] The second pulse is shifted in time to record the phase of (d) 0.8 the oscillation [33], as shown in Fig. 2(d) (red curve). To measure the effect of the vacuum on the frequency, the sodium BEC is removed by a resonant light pulse before 0.6 the Ramsey sequence starts. This light pulse does not have any observable effect on the Li sample. We clearly observe a fringe shift [Fig. 2(d), blue curve] in the absence of the 0.4 BEC. The frequency change is given by Δω ¼ Δϕ=Δt, where Δt is the total evolution time, including the time

excited state fraction of the state coupling (pulses). The observed sign of the 0.2 phase shift Δϕ corresponds to an increase of the energy 0 0.2 0.4 0.6 0.8 difference. readout phase For a quantitative comparison, care has to be taken since the employed near-resonance lattice for lithium FIG. 2. Motional Ramsey spectroscopy. (a) The sequence of also induces a weak modulation of the sodium BEC, as total time t consists of a π=2 pulse creating an equal superposition shown in the upper row of Fig. 3. In the Thomas-Fermi between the ground and the excited state and a subsequent free approximation, the modulation of the BEC density is ϕ π 2 evolution during which a phase is accumulated. A second = ρ ¼½μ − V ðxÞ=g , with g ¼ 4πℏ2a =m , where pulse maps the phase shift on the population η. (b) Shaking the mod V V V V V V ℏ ¼ h=2π, mV is the mass of the sodium atoms, and optical lattice couples the motional states of the lithium particles 54 54 and allows the preparation of a superposition of the two lowest aV ¼ . a0 is the intraspecies scattering length [34].It motional states. (c) Changing the evolution time t results in a modifies the effective confinement of the minority species Ramsey fringe in the population of the excited state, which is read due to mean-field interactions between the two species. out via band mapping on the first and second Brillouin zone. These interactions lead to the additional effective potential Typical optical density (OD) profiles of the resulting absorption for the particles. The resulting potential for the lithium images are depicted with a field of view of 1176 μm. The dashed atoms is, therefore, a combination of the optical lattice − ρ lines mark the limit of the Brillouin zones. (d) An exemplary and the modulated BEC, VP;effðxÞ¼VPðxÞ gPV mod, with interference fringe of 6Li after an evolution time of 1.1 ms is 2 gPV ¼ 2πℏ aPV=mr, mr is the reduced mass of the system; shown. The green and the blue dot correspond to the profiles that see, e.g., Refs. [23,35–37]. The band gap between the first are presented above. In the case of an evolution with the BEC and excited band in the effective potential is ω . Within present, the amplitude is reduced due to dephasing and the fringe eff is shifted, showing an increased energy difference. The shaded the approximation that the sites are independent harmonic area corresponds to the 1σ confidence levels of the phase oscillators, the relative shift due to the lattice is then estimation. ω − ω 1 δ eff 0 ≃ − gPV VV latt ¼ : ð1Þ accumulates a phase with respect to the ground state ω0 2 gV VP corresponding to its energy difference. The accumulated phase difference can be mapped onto an observable population difference by applying an addi- These effects are isolated by performing the Ramsey tional π=2 pulse. In our system of periodically arranged spectroscopy at different detunings of the lattice, as shown harmonic oscillators, the population difference can be in the lower left-hand panel of Fig. 3. While going closer accessed by the band-mapping technique, where the to the lithium resonance, the potential depth VP is kept quasimomentum in the lattice is mapped onto the free constant by reducing the light intensity accordingly. expansion of an atomic cloud [30–32]. While adiabatically Since the transition frequency for the background BEC reducing the lattice depth in 2 ms with a time constant is far detuned, the corresponding potential VV is reduced τ ¼ 0.8 ms, the longitudinal confinement through one and the modulation suppressed. We observe that the beam of the ODT is turned off, allowing the atoms to frequency shifts have a finite offset at the interpolated δ expand in an optical waveguide along the lattice direction. limit of zero modulation, which we denote with self. The After 10 ms of evolution an absorption image is taken. With fit yields for fermionic 6Li (blue dots) the relative shift

041041-3 T. RENTROP et al. PHYS. REV. X 6, 041041 (2016) X fermionic impuritiesmodulated bosonic impurities † BEC HP ¼ Ekaˆ kaˆ k; VP Xk ˆ † ˆ HV ¼ ωqbqbq; V q V X ˆ † ˆ ˆ ˆ † HInt ¼ Vqakþqakðbq þ b−qÞ: ð3Þ x 10-3 x 10-3 k;q≠0 8 8

6 6 Ek represent the energy levels of thep uncoupledffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi particles. ω 1 ξ 2 2 1=2 ρ 0 F 0 q ¼ cq½ þð qÞ = , where c ¼ gV =mV is the 4 self 4 BEC / 2 / 2 B speed of sound, describes the dispersion relation of the self excitations to the quantum vacuum. The particle (vacuum 0 0 excitation) creation and annihilation operators are denoted -2 -2 † ˆ † ˆ aˆ k (bq) and aˆ k (bq). The third term arises from the density- -4 -4 density interaction between the particles and the BEC. It -0.010 0.01 -0.010 0.01 V /V V /V describes the change in momentum of the particle via the V P V P absorption or emission of a phonon. We emphasize that the FIG. 3. Observation of the energy shift. The residual modula- coupling strength Vq captures the contact interaction in our tion of the density of the background BEC leads to an additional system, which is different from the long-range Coulomb energy shift of the particle energy level. This can be identified by interaction of an electron in a semiconductor and which is altering the ratio VV =VP through a change of the detuning of the given by optical lattice while keeping the potential depth for the particle pffiffiffiffiffiffiffiffiffiffi constant. A clear linear dependence on VV =VP is observed. ρ ξ 2 ξ 2 2 1=4 Vq ¼ gPV BECfð qÞ =½ð qÞ þ g ; ð4Þ Interpolating the measurement data to VV ¼ 0 allows for ex- traction of the energy shift. The positive offset is a direct signature ξ 8πρ −1=2 of the phononic Lamb shift, since it contradicts the negative where ¼ð BECaVÞ is the healing length of the ρ ξ energy shift expected from the increased effective mass. The condensate. Given the varying density BEC in the trap shaded areas show the theoretically predicted shift in the Fröhlich varies and has to be recalculated for each specific case, but scenario, taking into account the uncertainties of the experimental it is typically on the order of 0.5 μm. In the case of weak parameters. We find perfect agreement without free parameters interactions, when the interaction parameter, for fermionic particles (blue). In the case of a noncondensed bosonic particle cloud (green squares), no shift is observable in 2 2 2 α aPV mr EPV agreement with the theoretical expectation. For a condensed ¼ ¼ 2 2 ; ð5Þ ξa πm E bosonic particle cloud (red diamonds), we observe a significant V V Ph change of the energy shift. The theoretical expectation, not taking α ≪ 1 into account thermal excitations (T ¼ 0), is depicted as the red is much smaller than unity, , the particle-phonon shaded area. interaction term allows for perturbative treatment. Here, we introduce the characteristicp energyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi scale for ρ ξ−3 particle-phonon interaction, EPV ¼ gPV BEC , and the characteristic energy of the phonons, E ℏc=ξ . So, δF 6 4 1 2 10−4 Ph ¼ð Þ self ¼ . ð . Þ × , where the error bar corresponds to according to Refs. [23,38], α ≪ 1 simply states that the the 68% confidence interval. coupling should be small enough to avoid the quantum This positive shift corresponds to an increase of the depletion of the condensate. energy difference, which is at odds with a naive interpre- The canonical interpretation of the lowest-order correc- tation of an increased mass, and a more precise theoretical tion to the particle dispersion is that of an enhanced treatment including the phononic Lamb shift is necessary. effective mass mP >mP. Following the lines of Ref. [39] by calculating the lowest-order self-energy diagram (on IV. THEORETICAL TREATMENT shell) and computing its derivative with respect to squared 1 να momentum, we find mP=mP ¼ þ þ, where A commonly used model describing strongly imbal- ν ≈ 0.364 and 0.336 for 6Li and 7Li setups, respectively. anced mixtures is the celebrated Fröhlich Hamiltonian These are the results for the unconstrained system, when [23,24]: there is no confinement potential for both the BEC as well as the particles. Should the fermions be confined in a harmonic potential with frequency ω0, then a naive H ¼ HP þ HV þ H ; ð2Þ Int expectation for the shift of the energy levels is ω0 → ω0ð1 − να=2 þÞ. This entails a downward energy where the individual components are given by level shift corresponding to a slow down of the oscillations

041041-4 OBSERVATION OF THE PHONONIC LAMB SHIFT WITH A … PHYS. REV. X 6, 041041 (2016) and growing with the primary quantum number [40]; see where φnðxÞ are the wave functions of the nth eigenstate. Fig. 1(a). Finally, the Hamiltonian of the phonons is a continuum This is not supported by the experimental evidence, version of Eq. (3): and one must improve the model. In order to include this ’ Z effect, we take care of the system s geometry by modifying 3q the Hamiltonian Eq. (2). From now on, we consider d ω ˆ † ˆ HV ¼ 3 qbqbq: ð9Þ the particles as being confined in a parabolic potential ð2πÞ ℏω (energypffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi parameter 0 and length parameter aHO ¼ ℏ ω ≈ 0 24 μ =mP 0 . m) in the x direction, and being free Matrix elements for the particle-phonon scattering are still in all other spatial dimensions. Their eigenstates then have given by Eq. (4). 2 2 energies En;k ¼ ℏω0ðn þ 1=2Þþℏ k =2mP − μ, where k Here, the dimensionless interaction parameter is modi- is a 2D wave vector and n denotes the respective principal fied as compared to the free unconstrained system [see quantum number for the particles in the confinement Eq. (5)]: potential. The unperturbed particle Hamiltonian is then Z 2 ρ 1 2 X 2k gPV BECmV EPV d † α ¼ ¼ pffiffiffi : ð10Þ H ¼ E aˆ kaˆ k: ð6Þ con 8π2ℏ3ω ξ 2 ℏω P 2π 2 n;k n; n; 0 28π EPh 0 n ð Þ

Adapting the interaction term to the present setup leads to It can be interpreted as the ratio between the energy the following expression: shift of the dressed states and the energy spacing of the Z Z bare states. For our experimental parameters, α ≈ 10−4, 2 3 X con d k d q which justifies a perturbative treatment also in the con- H ¼ VqAðn1;n2;q Þ Int 2π 2 2π 3 x fined case. ð Þ ð Þ n1;n2 † ˆ ˆ † Our goal is the energy level renormalization for the × aˆ 0 aˆ k bq b−q : 7 n1;kþq n2; ð þ Þ ð Þ particles in the confinement potential. Just as in the case of the effective mass, its computation is best q0 Here, ¼ðqy;qzÞ denotes the transverse component of accomplished via lowest-order irreducible self-energy the phonon momentum. A is the matrix element for the correction, which is, in general, different for different transition between the harmonic oscillator energy levels: oscillator states n. The energy level structure is given by Z the retarded self-energy, which is computed via analytical − φ φ iqxx continuation of its Matsubara counterpart, see, Aðn1;n2;qxÞ¼ dx n1 ðxÞ n2 ðxÞe ; ð8Þ e.g., Ref. [39],

Z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ρ 3q X ξ 2 X∞ igPV BEC d ð qÞ 0 Σðn;k;iΩÞ¼ A ðn;m;q ÞAðm;n;q ÞG½m;k−q ;iΩ−ðm−nÞω0 −iϵ D0ðiϵ ;qÞ; β ð2πÞ3 ðξqÞ2 þ2 x x j j iϵj m¼0 ð11Þ

where Ω is a particle and ϵj ¼ 2πj=β bosonic (phonons) fermionic as well as bosonic particles. Performing the Matsubara frequencies. β ¼ 1=kBT is the inverse temper- calculation in the assumption that only the two lowest-lying ρ ature, where kB is the Boltzmann constant, and BEC is the energy levels of the confining potential are equally popu- density of the sodium BEC. The necessary Green functions lated, we compute their shifts due to interactions, which we are call self-energy shifts, and subsequently extract the energy difference δ between them. We take into account the 2ω self ϵ q q Fermi distribution of the particles and find that in the D0ði j; Þ¼ 2 2 ; ð12Þ ðiϵjÞ − ωq relevant particle density region the energy shifts are only weakly dependent on the particles’ chemical potential μ 0 ξ G½m; k − q ;iΩ − ðm − nÞω0 − iϵj and the ratio aHO= , which is typically on the order of 1, 1 having the general form ¼ ; ð13Þ iΩ − ðm − nÞω0 − iϵj − Em;k−q0 Δω where D0 and G are the phonon and the particle Green δ α μ ξ η self ¼ ¼ confð ;aHO; ; Þ; ð14Þ function, respectively. G has the same shape for both ω0

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μ ξ η where fð ;aHO; ; Þ is a dimensionless factor, an explicit describe energy level corrections of confined particles expression for this quantity is bulky and can be evaluated coupled to phonon continua. only numerically. We stress that, in general, the function f explicitly depends on the population imbalance η between V. PHONONIC LAMB SHIFT FOR the ground and excited states of the confinement potential. BOSONIC PARTICLES Assuming T ¼ 0, taking the actual experimental parameters in the case of symmetric superposition (when both Our new experimental platform further allows us to oscillator states are equally populated), we obtain (for investigate the phononic Lamb shift with bosonic particles, details see the Appendix) a feature not available in other experiments. Calculating the phononic Lamb shift now for bosonic particles at T ¼ 0 leads to 0.94 fðμ;a ; ξ; η ¼ 1=2Þ¼−0.11 þ : ð15Þ HO a =ξ Δω HO δ α ρ ξ η self ¼ ¼ confð 2D;aHO; ; Þ; ð17Þ ω0 Using this result we can calculate the expectation value of the density-dependent phase shift at each point in the cloud. The band-mapping technique then yields the spatially 2 f ρ2 ;a ;ξ;η ρ2 ξ g a =ξ 1 − η − g a =ξ η ; averaged phase shift Δϕ. However, different parts of ð D HO Þ¼ D ½ gð HO Þð Þ eð HO Þ the lithium cloud are embedded in varying sodium ð18Þ densities, which leads to locally varying phase shifts, ϕ r ω δ r 2 ð Þ¼ 0 selfð Þt. The coupling of the lithium particles 8π ξ 2 g ða =ξÞ¼pffiffiffi eðaHO= Þ ½1 − erfða =ξÞ; ð19Þ to the large number of phononic modes is sufficiently g HO 2 HO strong that it leads to observable loss of visibility, through ffiffiffiffiffiffi the dephasing of the different modes, during the Ramsey aHO p g ða =ξÞ¼4π 2π sequence and, therefore, limits the resolution. It can be e HO ξ taken into account via a position-dependent relaxation rate ffiffiffi p aHO ξ 2 Γ r 1 − π ðaHO= Þ 1 − ξ ð Þ, which was studied in detail in Ref. [18]. The total × ξ e ½ erfðaHO= Þ : signal of the fermions is then given by ð20Þ R d3rρ ðrÞe−ΓðrÞt sin½ϕðrÞ Δϕ ¼ asin R Li : ð16Þ A crucial difference from the fermionic case is the F 3rρ r −ΓðrÞt d Lið Þe explicit occurrence of the two-dimensional density of ρ condensed particles 2D in Eq. (18). The physical reason The density distributions are obtained numerically by is that the condensed particles populate a single state and taking into account the finite temperature as well as the thus coherently scatter the phonons, leading to an ampli- external potentials, such as the optical lattice and the fication of the signal. Employing bosonic particles thus harmonic trapping due to the ODT. For lithium it is results in a larger Lamb shift due to bosonic enhancement. necessary to include the additional potential due to the We use this feature to boost the Lamb shift in experi- interaction with sodium. We can then calculate the fringe ments with bosonic particles (7Li) that are condensed shift according to Eq. (16), as shown by the shaded inset in (≈60% condensate fraction). The results (red diamonds) Fig. 3, left-hand panel. The predicted phase shift is in very are displayed in Fig. 3. We observe that the effect of the good agreement with our measurements. modulation of the background BEC is inverted and smaller The conventional notion of the Lamb shift is the energy than for fermionic particles, as the interspecies scattering difference between the s and p electron states in a hydrogen length changes sign and is reduced by a factor of 3. Thus, atom. It is, however, computed with the renormalized the energy shift for a single boson is predicted to be electron mass as the coupling amplitude of the quantum 10 times smaller than for fermionic 6Li. Nevertheless, the electrodynamics (fine-structure constant) cannot be observed energy shift is amplified by a few thousands of switched off. According to this orthodox picture, the true bosons in the ground state of each pancake, leading to δB 4 1 1 10−3 Lamb shift in our system thus amounts to the energy self ¼ . ð Þ × . For comparison, we also perform difference between the shift due to the effective mass the same experiments with a noncondensed cloud of 7Li increase and the self-energy shift for confined particles; see atoms (green squares) at a temperature of ≈450 nK, leading Fig. 1(a). A direct measurement of the effective mass for to smaller numbers of atoms per quantum state, and a particles in a phononic vacuum along the lines of Ref. [15] weakened coupling as we operated at an approximately could help to clarify this subtlety. However, this important 2 times smaller BEC density. As expected from the point is often ignored in the solid-state and semiconductor theoretical predictions, we do not observe any shift in this context, and the term Lamb shift is generally used to regime.

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x 10-3 The observed total phase shift for the condensed particles 10 can then be calculated from Eq. (17) as a lithium density weighted mean: 8 R d3rρ ðrÞ sin½ϕðrÞ Δϕ R Li;BEC : 6 B ¼ asin 3rρ r ð21Þ d Li;BECð Þ 4 Here, we assume that only the condensed part experiences an energy shift, as the thermal part will not experience the 2 Bose amplification and the phononic Lamb shift is therefore negligible. However, the total signal is given as a 0 weighted sum over shifted and unshifted phase patterns. The weak interaction between the BEC and the bosonic -2 τ 50 particles leads to long coherence times ( > ms), which 0 0.2 0.4 0.6 0.8 1 are much longer than the Ramsey sequence, such that decoherence can be neglected in this case. In order to capture properly the lithium occupation number in the FIG. 4. Population dependence of the energy shift. The energy individual sites, we slice the ODT density according to the shift depends on the relative state population η as ground and periodicity of the lattice. These occupation numbers enter excited state experience different energy shifts. This can be the detailed calculation of the density distribution in the observed in a Ramsey sequence with an unequal superposition during free time evolution, keeping the other relevant parameters total confinement potential, where a 2D description is ρ constant. Experiments with bosonic particles are shown by red applied, and 2D is obtained for each pancake individually. η 0 81 7 diamonds. From these data we extract a critical 0;expt ¼ . ð Þ, The theoretically predicted phase shift is shown by the η in good agreement with theoretical predictions of 0;theor ¼ red shaded area in the right-hand panel of Fig. 3. It is much 0.85ð2Þ. For fermionic particles (blue circles) we do not observe smaller than the observed shift. However, a quantitative any dependence on η, as expected from our calculations. comparison between theory and experiment is difficult in this specific case. The overlap between the two species is much more sensitive to details in the trap geometries, By varying the length of the first Ramsey pulse, the 6 relative occupation of the excited state during time evolu- limiting its control. For the fermionic Li, Pauli blocking δ stabilizes the density distribution and the attractive inter- tion is changed. Figure 4 depicts the dependence of self actions with the condensate lead to a large overlap between on the excited fraction and confirms the prediction [42]. the two clouds. On the other hand, in the bosonic case, The experiments with fermions (blue) do not reveal any the condensation dramatically changes the density profile significant change. This is in agreement with the theoretical for temperatures of the order of the critical one. Further, analysis of the result [Eq. (14)], which reveals only weak 7 dependence on η below the experimental sensitivity of our the repulsive interactions between the bosonic Li and the setup. In contrast, the data with bosonic particles (red) show condensate push the minority species to the edge of the a clear linear dependence in agreement with the theoretical sample, where the finite-temperature effects are most prediction of Eq. (18). Motivated by this observation, we pronounced. This poses a challenge to the theoretical extract the crossing point via a linear fit, where the slope description. In addition, despite the fact that the interpar- and the offset are left as free parameters. The observed ticle interaction is very weak, we cannot exclude its crossing point at η0 ¼ 0.81ð7Þ is in excellent agreement contribution to the discrepancy. ;expt with the theoretical prediction. This observation confirms For a more quantitative comparison between theory and our understanding of the phononic Lamb shift for bosonic experiment for bosonic particles, we measure the energy particles within the developed frameworks. It also suggests shift as a function of different populations of ground and that the disagreement in the amplitude of the phononic excited state. Since the energy shift is proportional to the Lamb shift might be due to uncontrolled systematic shifts occupation number for bosons, there exists a specific in the overlap between the BEC and the lithium particles, relative population η0, where the energy shifts are equal which should be investigated in future works. for both levels. Mathematically it corresponds to the point ρ ξ η 0 where fð 2D;aHO; ; 0Þ¼ . For our experimental setup, η 0 85 2 VI. CONCLUSION this implies a vanishing phase shift for 0;theor ¼ . ð Þ, where the uncertainty arises from the determination of the Our results are in the realm of implementation of analog density profile. We emphasize that this crossing point quantum simulators on the basis of ultracold gas mixtures depends only weakly on the overlap between the particles [32,43]. We open up an avenue for benchmarking a new and the BEC. As such, it puts the theoretical predictions to a class of many-body theories, comprising not only different precise test. mutually interacting particles, but also interactions of their

041041-7 T. RENTROP et al. PHYS. REV. X 6, 041041 (2016) collective excitations. Such models are, e.g., proposed to ExtreMe Matter Institute, and the European Commission play a decisive role in fundamental open problems as FET-Proactive grant AQuS (Project No. 640800). This the origin of the high-temperature superconductivity; see work is part of and supported by the DFG Collaborative Ref. [44]. Furthermore, this setup captures several impor- Research Centre “SFB 1225 (ISOQUANT)”.F.A.O. tant nonrelativistic features of QED. In this way, we realize acknowledges support by the IMPRS-QD. A. K. is sup- a new experimental platform for quantitative exploration ported by the Heisenberg Programme of the Deutsche of such fascinating phenomena such as Casimir effect in Forschungsgemeinschaft (Germany) under Grant No. KO low-dimensional systems, where quantum fluctuations are 2235/5-1. dramatically enhanced [45,46], as well as in dynamical – situations [47 49]. By varying the distance between the Note added.—Recently, we became aware of the recent lattice valleys, we could directly test the dependence of the observation of the Bose polaron in the strong coupling limit energy shift on this spacing and therefore measure sign and in the vicinity of a Feshbach resonance [51,52]. amplitude of the vacuum-induced forces. In a many-particle limit, the background-mediated interaction between the particles can also lead to a generation and detection of APPENDIX: DETAILS OF THE CALCULATIONS nonlocal entanglement [50]. Here, we present the details of the calculations in the ACKNOWLEDGMENTS fermionic case. First, we can compute the sum over the Matsubara frequencies ϵj in Eq. (11) and perform We would like to thank F. Grusdt, E. Demler, and the analytic continuation procedure iΩ → ω þ iδ in order R. Schmidt for discussions. This work was supported by to obtain the retarded self-energy (δ is a positive the Heidelberg Center for Quantum Dynamics, the infinitesimal):

Z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 X∞ d q ðξqÞ Σðn; k; ωÞ¼−g2 ρ Aðn; m; q ÞAðm; n; q Þ PV BEC ð2πÞ3 ðξqÞ2 þ 2 x x m¼0 Nq þ n Nq þ 1 − n × F þ F ; ω − ðm − nÞω0 − Em;k−q0 þ ωq þ iδ ω − ðm − nÞω0 − Em;k−q0 − ωq þ iδ where nF is the Fermi distribution function of particles and Nq ¼ 1=½expðβωqÞ − 1 stands for the Bose distribution of phonons. For the particles in the ground state of the confinement potential n ¼ 0, there two classes of processes: scatterings within the n ¼ 0 state and scatterings into the first excited state m ¼ 1. The corresponding terms are very similar and we exemplarily show the computations due to the first kind of transitions. In the above result it is convenient to go over to an integral over s ¼jk − q0j and over the angle θ between the vectors k and q0. After performing the latter, we obtain Z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2 ρ ξ 2 kþq0 1 − g BEC ð qÞ − 2 2 Nq þ nFðsÞ Nq þ nFðsÞ Σ k; ω PV 0 ðaHOqxÞ = 00ð Þ¼ 2 dqxdq 2 e sds þ ; ðA1Þ 8π k ðξqÞ þ 2 jk−q0j ω − E0;s þ ωq þ iδ ω − E0;s − ωq þ iδ where the subscript 00 reflects the fact that we consider particle scatterings within the ground state of the harmonic oscillator. In the next step, we use the Plemelj formula in order to split off the real part of the self-energy yielding the energy shift. This procedure formally amounts to taking the principal value of the above integrals in the limit δ → 0. Then 2 the energy correction for the particle with the energy ω ¼ k =2mP takes the following form: Z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ρ η02 η2 g BECmP þ x − ξ 2η2 2 δϵ PV η η0 ðaHO= Þ x= 00 ¼ 2 2 d xd 02 2 e 8π ℏ ξκ η þ ηx þ 2 Z 0 2 ðκþη Þ Nq n ϵ Nq 1 − n ϵ ϵ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ pFðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFð Þ × d 2 2 2 2 2 þ 2 2 2 2 2 ; ðA2Þ κ−η0 2 0 0 0 0 ð Þ κ − ϵ þ Y η þ ηx 1 þðη þ ηxÞ=2 κ − ϵ − Y η þ ηx 1 þðη þ ηxÞ=2 pffiffiffi η0 0ξ η ξ κ ξ 2 where the integrals are over dimensionless variables ¼ q , x ¼ qx , ¼ k , and Y ¼ mP=mV. Here, one has to keep in 2 2 mind that the fermionic chemical potential μ is now measured in units of ℏ =ð2mPξ Þ. The remaining integrals can be 2 2 evaluated only numerically. The result depends on μ2mPξ =ℏ (which is ultimately related to the band imbalance parameter η; ξ κ 0 see main text), the ratio aHO= , and . It turns out that, in the relevant experimental parameter regime (T ¼ and numbers

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