Exact quantum field mappings between different experiments on quantum gases

Experiments on trapped quantum gases can probe challenging regimes of quantum many-body dynamics, where strong interactions or nonequilibrium states prevent exact solutions. Here we present a different kind of exact result, which applies even in the absence of actual solutions: a class of space-time mappings of different experiments onto each other. Since our result is an identity relating second-quantized field operators in the Heisenberg picture of quantum mechanics, it is extremely general; it applies to arbitrary measurements on any mixtures of Bose or Fermi gases, in arbitrary initial states. It represents a strong prediction of quantum field theory which can be tested in current laboratories, and whose practical applications include perfect simulation of interesting experiments with other experiments which may be easier to perform.

Figure 1

Space-time mapping between expectation values of an arbitrary observable $\widehat{O}$ in different experiments. The mapping consists of a time-dependent dilatation of space and multiplication of field operators by a Gaussian phase factor, and it relates observables at different times in the two experiments, which may be very different procedures. A might for example be free expansion after turning off the trap, while B is ramping to a Feshbach resonance. Observables are the same in the two experiments initially, because the gas is prepared in the same (arbitrary) state.

Figure 2

Numerical results for Gross-Pitaevski mean-field evolution showing densities ${\left|\psi (x,t)\right|}^2$ (A) and ${\left|\mathrm{\Psi }(x,t)\right|}^2$ (B), evolving under Eq. (3) with the operator fields replaced by complex classical fields, and a repulsive contact interaction whose initial strength is $U(x,x^{\prime },0)=g\left(0\right)\delta (x-x^{\prime })$ for $g\left(0\right)=50\hslash \omega a_0$ where $\omega$ is the initial trap frequency and $a_0=\sqrt{\hslash /\left(M\omega \right)}$ is the corresponding ground-state width. The integrated densities are normalized to 1; time and space are shown in units of $1/\omega$ and $a_0$, respectively. Note the different ranges of space and time covered by the two plots A and B, corresponding to Experiments A [expansion with $V_n=0, g\left(t\right)=g\left(0\right)$] and B [ramped interaction with constant trap, $g\left(t\right)=g\left(0\right)sec\left(\omega t\right)$], as described in the text. The space-time transformation maps the two plots onto each other. Even when mean-field theory is not valid, the quantum field mapping remains exact; this figure illustrates how it can relate nontrivial experiments. Here the initial state contains a dark soliton, which moves and changes in width while the whole cloud expands, and demonstrates how adiabaticity can break down at different times on different length scales. The dashed white curves in the B plot show the adiabatic Thomas-Fermi radius $R\left(t\right)=R\left(0\right){[cos\left(\omega t\right)]}^{-1/3}$.

Figure 3

Space-time evolution of the densities ${\left|\psi (x,t)\right|}^2$ and ${\left|\mathrm{\Psi }(x,t)\right|}^2$, respectively, plotted by mapping the densities obtained in experiments A (above) and B (below), according to Eq. (4) and its inverse. Since the mapping is exact, the barely visible differences between these plots, and those of Fig. 2 in our main text, are due to large rescalings of numerical solutions with finite resolution.

Figure 4

Time dependence in trap units $\hslash \omega a_0$ of the actual contact interaction strength $g\left(t\right)=g\left(0\right)sec\left(\omega t\right)$ (solid curve) and of two effective interaction strengths $g_{\mathrm{sol}}\left(t\right)$ (dash-dotted curve) and $g_{\mathrm{TF}}\left(t\right)$ (dotted curve), obtained by fitting small- and large-scale features of the density profile, respectively, as explained in the Appendix text. The effective interaction $g_{\mathrm{TF}}\left(t\right)$ represents the constant interaction strength for which the instantaneous parabolic envelope of the density profile would be a Thomas-Fermi ground state of the form of Eq. (B1). The effective interaction $g_{\mathrm{sol}}\left(t\right)$ represents the constant interaction strength for which the part of $\mathrm{\Psi }(x,t)$ near the soliton would be a gray soliton solution to the integrable nonlinear Schrödinger equation obtained by approximating the harmonic potential as constant. The fact that $g_{\mathrm{sol}}\left(t\right)$ follows $g\left(t\right)$ more closely than $g_{\mathrm{TF}}\left(t\right)$ shows that the condensate remains locally adiabatic on short length scales for longer than it remains adiabatic globally.