T ³-Interferometer for atoms

Abstract

The quantum mechanical propagator of a massive particle in a linear gravitational potential derived already in 1927 by Kennard [2, 3] contains a phase that scales with the third power of the time T during which the particle experiences the corresponding force. Since in conventional atom interferometers the internal atomic states are all exposed to the same acceleration a, this $T^3$-phase cancels out and the interferometer phase scales as $T^2$. In contrast, by applying an external magnetic field we prepare two different accelerations $a_1$ and $a_2$ for two internal states of the atom, which translate themselves into two different cubic phases and the resulting interferometer phase scales as $T^3$. We present the theoretical background for, and summarize our progress towards experimentally realizing such a novel atom interferometer.

Appendices

Appendix A: Dependence of $T^3$-phase on initial wave function

Fig. 6

Dependence of the numerical factor $\alpha =\alpha (\eta )$ defined by Eq. (36) on the dimensionless ratio $\eta$ of the time difference $\tau$ and spreading time $\tau _s$ given by Eqs. (10) and (34), respectively. For $\eta \rightarrow 0$ and $\eta \rightarrow \infty$ the factor $\alpha$ is almost constant and given by 1 / 6 and 1 / 24 (dashed line), respectively. However, for values of $\eta$ between these extremes $\alpha$ changes rapidly and thus in this transition domain the phase $\tilde{\phi }=\tilde{\phi }(\tau )$, Eq. (35), is not strictly cubic.

According to Sect. 2.1 the cubic phase $\phi$ in the propagator of a quantum particle moving in a linear potential manifests itself in every wave function exposed to this situation. Indeed, since $\phi$ is independent of the initial coordinate $z_\mathrm {i}$ it can be factored out of the Huygens integral for matter waves, Eq. (1).

Nevertheless, the integration of the remaining parts of the propagator in combination with the initial wave function can change the dependence of the phase of the final wave function on the propagation time $\tau =t_f-t_i$. To bring out this fact most clearly we consider the normalized initial wave function

$$\begin{aligned} \psi (z_\mathrm {i},t_\mathrm {i})\equiv \frac{1}{\left( \sqrt{\pi }\Delta z_0\right) ^{1/2}}\exp \left( -\frac{z_\mathrm {i}^2}{2\Delta z_0^2}\right) \end{aligned}$$

(30)

in the form of a real-valued Gaussian of width $\Delta z_0$ and note that wave functions of this kind can be easily realized nowadays in an experiment with cold atoms in a harmonic potential trap.

When we substitute this expression into the Huygens integral of matter waves, Eq. (1), and use the expressions, Eqs. (3) and (7), for the propagator we arrive at the final wave function

$$\begin{aligned} \psi (z_\mathrm {f},t_\mathrm {f})=\frac{1}{\left[ \sqrt{\pi }\Delta z(\tau )\right] ^{1/2}} \exp \left\{ -\frac{\left( z_\mathrm {f}-\frac{F}{2m}\tau ^2\right) ^2}{2\left[ \Delta z(\tau )\right] ^2}+i\beta \right\} \end{aligned}$$

(31)

with the time-dependent width

$$\begin{aligned} \Delta z(\tau )\equiv \Delta z_0\sqrt{1+\frac{\tau ^2}{\tau _\mathrm{s}^2}} \end{aligned}$$

(32)

and phase

$$\begin{aligned} \beta (\tau )\equiv \frac{F z_\mathrm {f} \tau }{\hbar }+ \frac{\tau }{\tau _\mathrm{s}}\frac{\left( z_\mathrm {f}-\frac{F}{2m}\tau ^2\right) ^2}{2\left[ \Delta z(\tau )\right] ^2}- \frac{F^2\tau ^3}{6\hbar m}-\frac{1}{2}\arctan \left( \frac{\tau }{\tau _\mathrm{s}}\right) . \end{aligned}$$

(33)

Here

$$\begin{aligned} \tau _\mathrm{s}\equiv \frac{m\Delta z_0^2}{\hbar } \end{aligned}$$

(34)

denotes the spreading time of the wave packet.

We combine the terms in Eq. (33), which are determined by the strength F of the force and independent of the final position $z_\mathrm {f}$, and find the total phase

$$\begin{aligned} \tilde{\phi }(\tau )\equiv -\alpha \left( \frac{\tau }{\tau _s}\right) \frac{F^2}{\hbar m}\,\tau ^3. \end{aligned}$$

(35)

Here we have introduced the time-dependent numerical factor

$$\begin{aligned} \alpha \left( \eta \right) \equiv \frac{1}{24} \frac{\eta ^2+4}{\eta ^2+1} \end{aligned}$$

(36)

depending on the dimensionless ratio $\eta$ of time difference $\tau$, Eq. (10), and spreading time $\tau _s$, which according to Eq. (34) is proportional to square of the initial width $\Delta z_0$.

For a plane wave we find $\Delta z_0\rightarrow \infty$, and thus $\tau _s\rightarrow \infty$ leading us to

$$\begin{aligned} \alpha \left( \eta \rightarrow 0\right) = \frac{1}{6}\,. \end{aligned}$$

(37)

However, for an infinitely narrow Gaussian with $\Delta z_0\rightarrow 0$, and thus $\tau _s\rightarrow 0$, we obtain

$$\begin{aligned} \alpha \left( \eta \rightarrow \infty \right) = \frac{1}{24}\,. \end{aligned}$$

(38)

So far we have restricted ourselves to the extreme limits of $\eta$. Only in the domains where $\alpha$ is approximately constant do we find a pure cubic phase dependence on $\tau$. Indeed, between the extremes the time dependence is more complicated as expressed in Fig. 6.

Appendix B: Raman pulses: superpositions and exchanges

In this Appendix, we describe the population dynamics [36, 64, 65] of the two resonant atomic states driven by the Raman laser pulses. For this purpose we consider the interaction between a three-level atom and two laser pulses of the form

$$\begin{aligned} \mathbf{E}_{1}(z,t)\equiv {\varvec{\mathcal E}}_{1}(t)\cos \left( k_1 z-\omega _1t+\phi _1\right) \end{aligned}$$

and

$$\begin{aligned} \mathbf{E}_2(z,t)\equiv {\varvec{\mathcal E}}_{2}(t)\cos \left( k_2 z-\omega _2 t+\phi _2\right) , \end{aligned}$$

(39)

where ${\varvec{\mathcal E}}_j$, $k_j$, $\omega _j$, and $\phi _j$ with $j=1,2$ denote the time-dependent envelope, frequency, wave vector, and phase of the jth field, respectively.

The laser frequencies $\omega _1$ and $\omega _2$ are assumed to only drive the transitions $|g_1\rangle \longleftrightarrow |e\rangle$ and $|g_2\rangle \longleftrightarrow |e\rangle$, respectively. Moreover, we assume that the laser pulses are so short that the atom does not move significantly during the interaction. Therefore, the position of the center-of-mass of the atom is considered to be fixed during the laser pulses.

Within the rotating-wave approximation [81] and in the limit of far-detuned laser pulses with identical detunings, that is when the Rabi frequencies $\Omega _{j}(t)\equiv \mathbf{d}_{g_j e}\cdot {\varvec{\mathcal E}}_j(t)/\hbar$ of the transitions $|g_j\rangle \longleftrightarrow |e\rangle$ are much smaller than the detuning $\Delta _j\equiv \omega _j-\omega _{eg_j}$ of the two laser pulses, $|\Delta _j|\gg |\Omega _j|$, and $\Delta \equiv \Delta _1=\Delta _2$, we can eliminate the excited state $|e\rangle$ and neglect the Stark shifts $|\Omega _{j}(t)|^2/(4\Delta )$. The resulting effective Hamiltonian [36]

$$\begin{aligned}\hat{H}_{\mathrm {p}}&=\hbar \frac{\Omega _1(t)\Omega _2(t)}{4\Delta }\left( e^{i[\Delta k z+\phi _\mathrm {L}]}|g_1\rangle \langle g_2|\right. \nonumber \\& \quad \left. +\, e^{-i[\Delta k z+\phi _\mathrm {L}]}|g_2\rangle \langle g_1|\right) \end{aligned}$$

(40)

describes the transitions between the states $|g_1\rangle$ and $|g_2\rangle$ due to the Raman pulses. Here $\mathbf{d}_{g_j e}\equiv \langle g_j|\mathbf{d}|e\rangle$ and $\omega _{eg_j}\equiv (E_e-E_{g_j})/\hbar$ are the dipole moment matrix element and the frequency of the transition $|g_j\rangle \longleftrightarrow |e\rangle$, respectively, with $\Delta k\equiv k_2-k_1$ and the slowly varying laser phase $\phi _\mathrm {L}(t)\equiv \phi _2-\phi _1$.

To avoid a momentum transfer during the Raman transitions, we assume that the laser pulses propagate in the same directions along the z-axis, Eq. (39), and that the difference $\Delta k$ of the two wave vectors is small compared to the size $\delta z$ of the atomic wave packet, that is $|\Delta k| \delta z\ll 1$. In this case the dependence on z in Eq. (40) can be neglected and we arrive at

$$\begin{aligned} \hat{H}_{\mathrm {p}}\cong \hbar \frac{\Omega _1(t)\Omega _2(t)}{4\Delta }\left( e^{i\phi _\mathrm {L}} |g_1\rangle \langle g_2|+ e^{-i\phi _\mathrm {L}}|g_2\rangle \langle g_1|\right) . \end{aligned}$$

(41)

The interaction of the atom with the two far-detuned Raman pulses, corresponding to Eq. (39), during the time interval $t_\mathrm {i}<t<t_\mathrm {f}$, and with $\Omega _j(t_{\mathrm {i}})=\Omega _j(t_{\mathrm {f}})=0$, is given by the evolution operator [36]

$$\begin{aligned}\hat{U}_\mathrm {p}&\equiv {1}+\left( -\frac{i}{\hbar }\right) \int \limits _{t_\mathrm {i}}^{t_\mathrm {f}}\mathrm{{d}}t \hat{H}_{\mathrm {p}}(t)\nonumber \\&\quad+\left( -\frac{i}{\hbar }\right) ^2\int \limits _{t_\mathrm {i}}^{t_\mathrm {f}} \mathrm{{d}}t\int \limits _{t_\mathrm {i}}^{t}\mathrm{{d}}t'\hat{H}_{\mathrm {p}}(t) \hat{H}_{\mathrm {p}}(t')+\ldots \;, \end{aligned}$$

(42)

which can be expressed as

$$\begin{aligned}\hat{U}_\mathrm {p}(\theta )&=\left( \,|g_1\rangle \langle g_1|+|g_2\rangle \langle g_2|\,\right) \cos \left( \frac{\theta }{2}\right) \nonumber \\ &\quad- i\left( e^{i\phi _\mathrm {L}}|g_1\rangle \langle g_2|+e^{-i\phi _\mathrm {L}}|g_2\rangle \langle g_1|\right) \sin \left( \frac{\theta }{2}\right) , \end{aligned}$$

(43)

where

$$\begin{aligned} \theta \equiv \frac{1}{2\Delta }\int \limits _{t_\mathrm {i}}^{t_\mathrm {f}} \mathrm{{d}}t\,\Omega _1(t)\Omega _2(t) \end{aligned}$$

(44)

denotes the total pulse area.

The case $\theta =\frac{\pi }{2}$, which is a $\frac{\pi }{2}$-pulse, gives rise [67] to the coherent superpositions

$$\begin{aligned} \hat{U}_\mathrm {p}\left( \frac{\pi }{4}\right) |g_1\rangle =\frac{1}{\sqrt{2}} \left( |g_1\rangle - ie^{-i\phi _\mathrm {L}}|g_2\rangle \right) \end{aligned}$$

and

$$\begin{aligned} \hat{U}_\mathrm {p}\left( \frac{\pi }{4}\right) |g_2\rangle = \frac{1}{\sqrt{2}}\left( |g_2\rangle -i e^{i\phi _\mathrm {L}}|g_1\rangle \right) . \end{aligned}$$

(45)

In contrast, the case $\theta =\pi$, known as a $\pi$-pulse describes an exchange

$$\begin{aligned} \hat{U}_\mathrm {p}\left( \frac{\pi }{2}\right) |g_1\rangle =-ie^{-i\phi _\mathrm {L}} |g_2\rangle \end{aligned}$$

and

$$\begin{aligned} \hat{U}_\mathrm {p}\left( \frac{\pi }{2}\right) |g_2\rangle =-ie^{i\phi _\mathrm {L}}|g_1\rangle \end{aligned}$$

(46)

of the level populations.

Appendix C: Interferometer: sequence of unitary operators

Unitary operators describe both the interaction of the atom with the four Raman pulses and the time evolution associated with the center-of-mass motion. In the present Appendix, we derive the complete quantum state of the atom consisting of the internal states as well as the center-of-mass in the two exit ports of our interferometer following the procedure outlined in Refs. [28, 36, 67].

The dynamics in our interferometer consists of the following steps:

• 1. Before the first $\frac{\pi }{2}$-pulse, at $t=t_0-\varepsilon$, the initial state

  $$\begin{aligned} |\Psi (t_0-\varepsilon )\rangle \equiv |g_1\rangle |\psi _0\rangle \end{aligned}$$

  (47)

  consists of the center-of-mass motion $|\psi _0\rangle$ and the internal state $|g_1\rangle$. Here and throughout this Appendix $\varepsilon$ is an infinitesimally small and positive number.

• 2. After the first $\frac{\pi }{2}$-pulse at $t=t_0+\varepsilon$, the state reads

  $$\begin{aligned}|\Psi (t_0+\varepsilon )\rangle &=\hat{U}_\mathrm {p}\left( \frac{\pi }{4}\right) |\Psi (t_0-\varepsilon )\rangle \nonumber \\&=\left( \frac{1}{\sqrt{2}}|g_1\rangle -\frac{i}{\sqrt{2}}e^{-i \phi _\mathrm {L}(t_0)}|g_2\rangle \right) |\psi _0\rangle , \end{aligned}$$

  (48)

  where we have used Eq. (45).

• 3. Before the first $\pi$-pulse at $t=t_1-\varepsilon$, we find

  $$\begin{aligned}|\Psi (t_1-\varepsilon )\rangle &=\hat{U}_a\left( t_1,t_0\right) |\Psi (t_0+\varepsilon )\rangle \nonumber \\&= \frac{1}{\sqrt{2}}|g_1\rangle \hat{U}_{a_1} (t_1,t_0)|\psi _0\rangle \nonumber \\&\quad - \frac{i}{\sqrt{2}}e^{-i\phi _\mathrm {L}(t_0)}|g_2\rangle \hat{U}_{a_2}(t_1,t_0)|\psi _0\rangle . \end{aligned}$$

  (49)

• 4. After the first $\pi$-pulse at $t=t_1+\varepsilon$, we obtain

  $$\begin{aligned}|\Psi (t_1+\varepsilon )\rangle &=\hat{U}_\mathrm {p}\left( \frac{\pi }{2}\right) |\Psi (t_1-\varepsilon )\rangle \nonumber \\&=-\frac{i}{\sqrt{2}}e^{-i\phi _\mathrm {L}(t_1)}|g_2\rangle \hat{U}_{a_1}(t_1,t_0)|\psi _0\rangle \nonumber \\&\quad -\frac{1}{\sqrt{2}}e^{-i[\phi _\mathrm {L}(t_0)-\phi _\mathrm {L}(t_1)]} |g_1\rangle \hat{U}_{a_2}(t_1,t_0)|\psi _0\rangle . \end{aligned}$$

  (50)

• 5. Before the second $\pi$-pulse at $t=t_2-\varepsilon$, the state takes the form

  $$\begin{aligned}|\Psi (t_2-\varepsilon )\rangle &=\hat{U}_a(t_2,t_1) |\Psi (t_1+\varepsilon )\rangle \nonumber \\&=-\frac{i}{\sqrt{2}}e^{-i\phi _\mathrm {L}(t_1)}|g_2\rangle \hat{U}_{a_2}(t_2,t_1)\hat{U}_{a_1}(t_1,t_0)|\psi _0\rangle \nonumber \\&\quad - \frac{1}{\sqrt{2}}e^{-i[\phi _\mathrm {L}(t_0)-\phi _\mathrm {L} (t_1)]}|g_1\rangle \hat{U}_{a_1}(t_2,t_1)\hat{U}_{a_2} (t_1,t_0)|\psi _0\rangle . \end{aligned}$$

  (51)

• 6. After the second $\pi$-pulse at $t=t_2+\varepsilon$, we arrive at the state

  $$\begin{aligned}|\Psi (t_2+\varepsilon )\rangle &=\hat{U}_\mathrm {p}\left( \frac{\pi }{2}\right) |\Psi (t_2-\varepsilon )\rangle \nonumber \\&=-\frac{1}{\sqrt{2}}e^{-i[\phi _\mathrm {L}(t_1)-\phi _\mathrm {L}(t_2)]} |g_1\rangle \hat{U}_{a_2}(t_2,t_1)\hat{U}_{a_1}(t_1,t_0)|\psi _0\rangle \nonumber \\&\quad+\frac{i}{\sqrt{2}}e^{-i[\phi _\mathrm {L}(t_0)-\phi _\mathrm {L}(t_1) +\phi _\mathrm {L}(t_2)]}\nonumber \\&\quad\times |g_2\rangle \hat{U}_{a_1}(t_2,t_1) \hat{U}_{a_2}(t_1,t_0)|\psi _0\rangle . \end{aligned}$$

  (52)

• 7. Before the second $\frac{\pi }{2}$-pulse at $t=t_3-\varepsilon$, the state reads

  $$\begin{aligned}|\Psi (t_3-\varepsilon )\rangle &=\hat{U}_a(t_3,t_2) |\Psi (t_2+\varepsilon )\rangle \nonumber \\&=-\frac{1}{\sqrt{2}}e^{-i[\phi _\mathrm {L}(t_1) -\phi _\mathrm {L}(t_2)]}\nonumber \\&\quad\times |g_1\rangle \hat{U}_{a_1}(t_3,t_2)\hat{U}_{a_2}(t_2,t_1) \hat{U}_{a_1}(t_1,t_0)|\psi _0\rangle \nonumber \\&\quad +\frac{i}{\sqrt{2}}e^{-i[\phi _\mathrm {L}(t_0)-\phi _\mathrm {L}(t_1)+ \phi _\mathrm {L}(t_2)]}\nonumber \\&\quad \times |g_2\rangle \hat{U}_{a_2}(t_3,t_2)\hat{U}_{a_1}(t_2,t_1) \hat{U}_{a_2}(t_1,t_0)|\psi _0\rangle . \end{aligned}$$

  (53)

• 8. Finally, after the second $\frac{\pi }{2}$-pulse at $t=t_3+\varepsilon$, we conclude with the state

  $$\begin{aligned}|\Psi (t_3+\varepsilon )\rangle &=\hat{U}_\mathrm {p}\left( \frac{\pi }{4}\right) |\Psi (t_3-\varepsilon )\rangle \nonumber \\&= \frac{1}{2}\,e^{-i[\phi _\mathrm {L}(t_1)-\phi _\mathrm {L}(t_2)]} |g_1\rangle \left( e^{-i\varphi _\mathrm {L}}\hat{U}_{\mathrm {u}} -\hat{U}_{\mathrm {l}}\right) |\psi _0\rangle \nonumber \\&\quad +\frac{i}{2}e^{-i[\phi _\mathrm {L}(t_1)-\phi _\mathrm {L}(t_2) +\phi _\mathrm {L}(t_3)]}|g_2\rangle \left( e^{-i\varphi _\mathrm {L}}\hat{U}_{\mathrm {u}}+\hat{U}_{\mathrm {l}}\right) |\psi _0\rangle , \end{aligned}$$

  (54)

  where

  $$\begin{aligned} \hat{U}_{\mathrm {l}}\equiv \hat{U}_{a_1}(t_3,t_2)\hat{U}_{a_2}(t_2,t_1) \hat{U}_{a_1}(t_1,t_0) \end{aligned}$$

  and

  $$\begin{aligned} \hat{U}_{\mathrm {u}}\equiv \hat{U}_{a_2}(t_3,t_2)\hat{U}_{a_1} (t_2,t_1)\hat{U}_{a_2}(t_1,t_0) \end{aligned}$$

  (55)

  are the unitary evolution operators associated with the center-of-mass motion for the lower and the upper paths of the interferometer shown in Fig. 2, and

  $$\begin{aligned} \varphi _\mathrm {L}\equiv \phi _\mathrm {L}(t_0)-2\phi _\mathrm {L}(t_1)+2\phi _\mathrm {L} (t_2)-\phi _\mathrm {L}(t_3) \end{aligned}$$

  (56)

  is the total phase resulting from the action of the four laser pulses.

Appendix D: Conditions for a closed $T^3$-interferometer

In the preceding Appendix, we have derived an expression for the complete quantum state of the atom in the exit ports of the interferometer. Here we have allowed arbitrary times for the interactions with the laser pulses. In the present Appendix, we choose these times in such a way as to maximize the contrast.

The probability $P_{g_1}$ to observe atoms in the ground state $|g_1\rangle$ after the action of the four Raman pulses at $t=t_3+\varepsilon$, follows from the quantum state $|\Psi (t_3+\varepsilon )\rangle$ given by Eq. (54) and contains the state

$$\begin{aligned} |\psi _{g_1}\rangle \equiv \langle g_1|\Psi (t_3+\varepsilon )\rangle \end{aligned}$$

(57)

of the center-of-mass motion of atom in $|g_1\rangle$. It takes the form

$$\begin{aligned} P_{g_1}\equiv \langle \psi _{g_1}|\psi _{g_1}\rangle =\frac{1}{2} \left[ 1-C\cos \left( \varphi _\mathrm {i}+\varphi _\mathrm {L}\right) \right] , \end{aligned}$$

(58)

where the contrast C and the phase $\varphi _\mathrm {i}$ of the interferometer are the modulus and the argument of the matrix element

$$\begin{aligned} \langle \psi _0|\hat{U}^{\dagger }_\mathrm {u} \hat{U}^{}_\mathrm {l}|\psi _0\rangle \equiv Ce^{i\varphi _\mathrm {i}}. \end{aligned}$$

(59)

We maximize C, that is we have $C=1$, when we close our interferometer. In this case $P_{g_1}$ given by Eq. (58) is independent of the initial velocity and position of the atom.

To close the interferometer we have to find the time intervals $t_{j+1,j}\equiv t_{j+1}-t_{j}$ with $j=0,1,2$ between the Raman pulses shown in Fig. 2, such that the final velocities $v_\mathrm {u}(t_3)$ and $v_\mathrm {l}(t_3)$, as well as the final positions $z_\mathrm {u}(t_3)$ and $z_\mathrm {l}(t_3)$ on the upper and lower paths of the interferometer are identical.

Indeed, for the velocity we derive the following formulae:

• (i) for the upper path

  $$\begin{aligned} v_0&\rightarrow v_\mathrm {u}(t_1)=v_0+a_2t_{10}\\&\rightarrow v_\mathrm {u}(t_2)=v_\mathrm {u}(t_1)+a_1t_{21}\\&\rightarrow v_\mathrm {u}(t_3)=v_\mathrm {u}(t_2)+a_2t_{32} \end{aligned}$$

• (ii) for the lower path

  $$\begin{aligned} v_0&\rightarrow v_\mathrm {l}(t_1)=v_0+a_1t_{10}\\&\rightarrow v_\mathrm {l}(t_2)=v_\mathrm {l}(t_1)+a_2t_{21}\\&\rightarrow v_\mathrm {l}(t_3)=v_\mathrm {l}(t_2)+a_1t_{32}. \end{aligned}$$

As a result, the interferometer is closed in velocity space, if $v_ \mathrm {u}(t_3)=v_\mathrm {l}(t_3)$, that is,

$$\begin{aligned}&v_0+a_2t_{10}+a_1t_{21}+a_2t_{32}\\ &=v_0+a_1t_{10}+a_2t_{21}+a_1t_{32}, \end{aligned}$$

or, equivalently,

$$\begin{aligned} t_{10}-t_{21}+t_{32}=0. \end{aligned}$$

(60)

As for the position, we obtain the following rather lengthy expressions:

• (i) for the upper path

  $$\begin{aligned} z_0&\rightarrow z_\mathrm {u}(t_1)=z_0+v_0t_{10}+\frac{1}{2}a_2 t_{10}^2\\&\rightarrow z_\mathrm {u}(t_2)=z_\mathrm {u}(t_1)+v_\mathrm {u}(t_1)t_{21} +\frac{1}{2}a_1 t_{21}^2\\&\rightarrow z_\mathrm {u}(t_3)=z_\mathrm {u}(t_2)+v_\mathrm {u}(t_2)t_{32} +\frac{1}{2}a_2 t_{32}^2\\&=z_0+v_0(t_{10}+t_{21}+t_{32})+\frac{1}{2}(a_2t_{10}^2 +a_1t_{21}^2+a_2t_{32}^2)\\&\quad+a_2t_{10}(t_{21}+t_{32})+a_1t_{21}t_{32}, \end{aligned}$$

• (ii) for the lower path

  $$\begin{aligned} z_0&\rightarrow z_\mathrm {l}(t_1)=z_0+v_0t_{10}+\frac{1}{2}a_1 t_{10}^2\\&\rightarrow z_\mathrm {l}(t_2)=z_\mathrm {l}(t_1)+v_\mathrm {l}(t_1)t_{21} +\frac{1}{2}a_2 t_{21}^2\\&\rightarrow z_\mathrm {l}(t_3)=z_\mathrm {l}(t_2)+v_\mathrm {l} (t_2)t_{32}+\frac{1}{2}a_1 t_{32}^2\\&=z_0+v_0(t_{10}+t_{21}+t_{32})+\frac{1}{2}(a_1t_{10}^2 +a_2t_{21}^2+a_1t_{32}^2)\\&+a_1t_{10}(t_{21}+t_{32})+a_2t_{21}t_{32}. \end{aligned}$$

As a result, the interferometer is closed in position space if $z_\mathrm {u}(t_3)=z_\mathrm {l}(t_3)$, that is,

$$\begin{aligned} t_{10}^2-t_{21}^2+t_{32}^2+2t_{10}(t_{21}+t_{32})-2t_{21}t_{32}=0. \end{aligned}$$

(61)

When we solve the system of the two algebraic Eqs. (60) and (61), for $t_{21}$ and $t_{32}$ in terms of $t_{10}$, we obtain

$$\begin{aligned} t_3-t_2=t_1-t_0=T\;\;\;\mathrm{and}\;\;\;t_2-t_1=2T. \end{aligned}$$

(62)

Hence, to close the interferometer, the four Raman pulses must be separated in time by T, 2T, and T as indicated in Fig. 2.

Appemdix E: Interferometer phase

In the preceding Appendix, we have used classical trajectories to find the separation $T-2T-T$ between the four Raman pulses leading to a closed interferometer. We now show that in this case the product $\hat{U}_\mathrm {u}^{\dag }\hat{U}_\mathrm {l}$ of the evolution operators $\hat{U}_\mathrm {l}$ and $\hat{U}_\mathrm {u}$ defined by Eq. (55) is proportional [28, 37, 67] to the identity operator, that is

$$\begin{aligned} \hat{U}_\mathrm {u}^{\dag }\hat{U}^{}_\mathrm {l}=e^{i \varphi _\mathrm {i}}\,{1}, \end{aligned}$$

(63)

where $\varphi _\mathrm{i}$ is the interferometer phase.

Therefore, a normalized state $|\psi _0\rangle$ leads by virtue of Eq. (59) to a perfect contrast, that is $C=1$, indicating that the interferometer is independent of $|\psi _0\rangle$. Moreover, this calculation provides us with an explicit expression for $\varphi _\mathrm{i}$.

To evaluate the evolution operator

$$\begin{aligned} \hat{U}^{}_\mathrm{l}\equiv \hat{U}_{a_1}(T)\hat{U}_{a_2}(2T)\hat{U}_{a_1}(T) \end{aligned}$$

(64)

for the lower path of our interferometer, shown in Fig. 2, we use the Baker–Campbell–Hausdorff and Zassenhaus formulas [85] to represent the operator $\hat{U}_{a}(T)$ given by Eq. (13) in the form of a product

$$\begin{aligned} \hat{U}_{a}(T)=\exp \left( i\frac{ma^2T^3}{12\hbar }\right) \hat{\mathcal {D}} \left( \frac{1}{2}aT^2,maT\right) \hat{U}_{0}(T) \end{aligned}$$

(65)

consisting of a phase factor, the displacement operator

$$\begin{aligned} \hat{\mathcal {D}}\left( Z,P\right) \equiv \exp \left[ -\frac{i}{\hbar } \left( Z\hat{p}_z-P\hat{z}\right) \right] , \end{aligned}$$

(66)

and the unitary operator

$$\begin{aligned} \hat{U}_{0}(T)\equiv \exp \left( -\frac{i}{\hbar }\frac{\hat{p}_z^2}{2m}T\right) \end{aligned}$$

(67)

of a free particle.

The decomposition, Eq. (65), allows us to rewrite Eq. (64) as

$$\begin{aligned} \hat{U}^{}_\mathrm {l}&=\exp \left[ i\frac{m(a_1^2+4a_2^2)}{6\hbar }T^3\right] \hat{\mathcal {D}}\left( \frac{1}{2}a_1 T^2,ma_1T\right) \hat{U}_{0}(T)\\&\quad\times \hat{\mathcal {D}}\left( 2a_2 T^2,2ma_2T\right) \hat{U}_{0}(2T) \hat{\mathcal {D}}\left( \frac{1}{2}a_1 T^2,ma_1T\right) \hat{U}_{0}(T). \end{aligned}$$

(68)

With the help of the commutation relation

$$\begin{aligned} \hat{U}_{0}(T)\hat{\mathcal {D}}\left( Z,P\right) =\hat{\mathcal {D}} \left( Z+\frac{P}{m}T,P\right) \hat{U}_{0}(T) \end{aligned}$$

and the addition identity

$$\begin{aligned} \hat{U}_{0}(T_1)\hat{U}_{0}(T_2)=\hat{U}_{0}(T_1+T_2) \end{aligned}$$

for the operators $\hat{\mathcal {D}}$ and $\hat{U}$ given by Eqs. (66) and (67), we can shift all free-evolution operators $\hat{U}_0$ in Eq. (68) to the right and we arrive at

$$\begin{aligned} \hat{U}^{}_\mathrm {l}=\exp \left[ i\frac{m(a_1^2+4a_2^2)}{6\hbar }T^3\right] \hat{\mathcal {D}}\left( \frac{1}{2}a_1 T^2,ma_1T\right) \end{aligned}$$

$$\begin{aligned} \quad \times \hat{\mathcal {D}}\left( 4a_2 T^2,2ma_2T\right) \hat{\mathcal {D}}\left( \frac{7}{2}a_1 T^2,ma_1T\right) \hat{U}_{0}(4T), \end{aligned}$$

or

$$\begin{aligned}\hat{U}^{}_\mathrm {l}&=\exp \left[ i\frac{m T^3}{3\hbar }\left( 5a_1^2+9a_1a_2+2a_2^2\right) \right]\\&\quad \times \hat{\mathcal {D}}\left( 4(a_1+a_2)T^2,2m(a_1+a_2)T\right) \hat{U}_{0}(4T). \end{aligned}$$

(69)

In the last step we have made use of the addition identity

$$\begin{aligned} \hat{\mathcal {D}}\left( Z_1,P_1\right) \hat{\mathcal {D}}\left( Z_2,P_2\right) =e^{i\tilde{\varphi }} \hat{\mathcal {D}}\left( Z_1+Z_2,P_1+P_2\right) \end{aligned}$$

with

$$\begin{aligned} \tilde{\varphi }\equiv \frac{1}{2\hbar }\left( P_1Z_2-P_2 Z_1\right) , \end{aligned}$$

to combine all three displacement operators into a single one.

Since the evolution operator $\hat{U}^{}_\mathrm {u}$ defined by Eq. (55) for the upper path of our interferometer follows directly from the operator $\hat{U}^{}_\mathrm {l}$ given by Eq. (69) for the lower path by an exchange of the accelerations $a_1$ and $a_2$, we arrive at

$$\begin{aligned} \hat{U}^{}_\mathrm {u}=\exp \left[ i\frac{m T^3}{3\hbar }\left( 2a_1^2+9a_1a_2+5a_2^2\right) \right] \end{aligned}$$

$$\begin{aligned} \times \,\hat{\mathcal {D}}\left( 4(a_1+a_2)T^2,2m(a_1+a_2)T\right) \hat{U}_{0}(4T). \end{aligned}$$

(70)

When we substitute Eqs. (69) and (70) into the left-hand side of Eq. (63) and use the property that the operators $\hat{\mathcal {D}}$ and $\hat{U}_{0}$ are unitary, the interferometer phase reads

$$\begin{aligned} \varphi _\mathrm {i}=\frac{m}{\hbar }\left( a_1^2-a_2^2\right) T^3. \end{aligned}$$

(71)

Hence, $\varphi _\mathrm{i}$ is independent of the initial position $z_0$ and velocity $v_0$ as well as of the initial state. Moreover, it scales with the third power of the time interval $T\equiv t_1-t_0$ between the first and the second Raman pulses.